Table of Contents
ACKNOWLEDGMENT ............................................................................................ ii
Describes a mathematical system (M8GE-IIIa-1) .................................................. 1
Illustrates the need for an axiomatic structure of a mathematical system in
general, and in Geometry in particular: (a)defined terms; (b)undefined terms;
(c)postulates; and (d)theorems (M8GE-IIIa-c-1) ................................................... 20
Illustrates triangle congruence (M8GE-IIId-1) ...................................................... 46
Illustrates the SAS, ASA and SSS congruence postulates (M8GE-IIId-e-1) ....... 64
Solves corresponding parts of congruent triangles (M8GE-IIIf-1) ........................ 76
Proves two triangles are congruent (M8GE-IIIg-1) ............................................... 90
Proves statements on triangle congruence (M8GE-IIIh-1) ................................. 105
Applies triangle congruence to construct perpendicular lines and angle bisectors
(M8GE-IIIi-j-1) ...................................................................................................... 132
Pre-Test and Post -Test
i
ACKNOWLEDGMENT
With deep appreciation and gratitude for the expertise and collaborative
efforts of various individuals as members of the Development Team on the writing,
editing, validating, and printing of the Contextualized Prototype Daily Lesson Plans
in Mathematics 8 (Third Quarter).
WRITERS
Flocerpida B. Barias
Romer B. Brofas
Ruben B. Boncocan Jr.
Maria Elvira R. Estevez
Charlie B. Maduro
Regine B. Bueno
Vicky B. Bermillo
Rigor B. Bueno I
Nancy A. Montealegre
Rowena B. Benoyo
Hilda J. Carlet
Sylvia B. Sariola
Nerissa A. Mortega
EDITORS AND VALIDATORS
Dioleta B. Borais
Hilda J. Carlet
Efleda C. Dolz
Nerissa A. Mortega
Aladino B. Bonavente
DEMONSTRATION TEACHERS
Jennifer B. Binasa
Rigor B. Bueno I
Jennylyn B. Cid
Analyn B. Lovendino
Nancy A. Montealagre
Vicky B. Bermillo
Emmalyn B. Manuel
Evany Cortezano
Aiko B. Adonis
Elsa B. Arevalo
Christian B. Barrameda
Carlos B. Borlagdan
Jocelyn Beren
LAY-OUT ARTIST
Marisol B. Boseo
Ruel Brondo
DIOLETA B. BORAIS
Education Program Supervisor, Mathematics
MARVIN C. CLARINA
Chief, Curriculum Implementation Division
BERNIE C. DESPABILADERO
Asst. Schools Division Superintendent
MARIANO B. DE GUZMAN
OIC, Schools Division Superintendent
ii
(M8GE-IIIa-1) - Describes a Mathematical System
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 1, Day 1
The learner demonstrates understanding of key
concepts of axiomatic structure of geometry.
The learner is able to formulate an organized plan to
handle a real – life situation.
The learner describes a mathematical system.
M8GE-IIIa-1
Describing a mathematical system (Undefined terms)
-
Geometry pp.3-4(Textbook for Third Year)
Advanced Learners
Activity 1
Average Learners
Activity 1
“YES OR NO?”
“DISCOVER ME!”
Direction:
Instruct
the Direction: Arrange the
students to show the YES jumbled letters to form a
card if they know the word or basic geometric word.
phrase. Show the NO card if
they are not familiar with the
1. IOPNT
word or phrase. Call a
2. NLIE
student showing a YES card
3. LPAEN
to share what he/she know
4. TRYMEOGE
about the word or phrase.
5. MATHCALTIMA
1. Geometry
E STEMSY
2. Point
3. Line
4. Plane
5. Mathematical system
1
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
Activity 2
Activity 2
DESCRIBE ME!
I WANNA BE COMPLETE!
Direction: Divide the
class into five groups.
Each group will
describe a geometric
word by completing the
statement within 2
minutes.
Output presentation
followed.
Direction: Complete the
table below by supplying
the missing word/term.
Picture/
Represent
-ation
dot
Straight
mark
with two
arrow
heads
A B
It is read as _____ AB.
Line AB is represented
by a straight mark with
________ arrow heads.
C ℬ
It is read as _____.
Plane ABC has infinite
length, has infinite ____
but has no ____.
2
By using
two
capital
letters
with a
double
arrowhea
d above
them or
lower
case
letter
Has
infinite
_____
Has
no__
Has no
____
____
𝑚
Read as
line AB or
line m
Slanted
four sided
figure
By using
single
capital script
letter or by
three___
𝑚
Line m has infinite
points. It has no ___ and
no ___.
B
Point
A B
Group 3: Given
A
By using Has no
a capital length
___
Has no _
Has no _
Undefined
term
A
Group 2: A
Point A has no length,
has no ______, and has
no _______.
Group 5: Given
Description
Read as
point A.
A
Group 1: Given
It is read as _____ A.
Point A is represented
by a _____.
Group 4: Given
HOW TO
Name?
Read as
plane ABC
or plane.
Has
infinite
length
Has
infinite__
_
Has
no___
It is a
___
surface
Plane__
___
2. Processing the
answer
3. Reinforcing the skills
Questions:
1. What are the undefined terms in geometry?
Why are they undefined?
2. How do you represent a point? a line? and a
plane?
3. How do you denote a point? a line? and a
plane?
4. How do you describe a point? a line?
and a plane?
5. Cite some real-world objects illustrating a point,
a line and a plane.
(The teacher will emphasize the representations
of a point, of a line and of a plane abound in
nature.)
A. Use the figure to answer each of the questions
below.
G
O
D
ℛ
1.
2.
3.
B.
What are the given points?
What is the name of the line?
What is the name of the plane?
Give the characteristics of the following undefined
terms represented by the following objects.
1. top of a box
2. side of a blackboard
3. tip of a pen
4. a corner of a room
5. cover of a book
4. Summarizing the
Lesson
C. Assessment:
D. Agreement/
Assignment:
How do you describe a point, a line, and a plane?
Describe the following undefined terms:
1. point
2. line
3. plane
Cut out some pictures that will show representations of
point, line, and plane.
V. REMARKS:
VI. REFLECTION:
VII. OTHERS
A. No. of learners who
earned 80% in the
evaluation
3
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
4
(M8GE-IIIa-1) - Describes a Mathematical System
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
a) Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 1, Day 2
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry.
The learner is able to formulate an organized plan to
handle a real – life situation.
The learner describes a mathematical system.
M8GE-IIIa-1
Describing a Mathematical System (Defined Terms)
Moving Ahead with Mathematics II pp.76-79
Advanced Learners
Average Learners
Activity 1
“HUNT ME!”
Direction: The array of letters below includes some basic
geometric words. Mark the words by circling them.
1.
2.
3.
4.
5.
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
BRAYT
TRSEGMENTS
ABCMIDPOINTH
BSKEWR
ARTCOLLINEARK
Activity 2
Activity 2
“DEFINE ME”
“MATCH ME!”
5
Direction: Divide the class
into ten groups. Each group
will be given a task to
complete the sentence
within 1-2 minutes.
Group 1: Given:
𝑌
In symbol: ⃗⃗⃗⃗⃗
𝑋𝑌
a) Read as _____.
b) A __ is a subset of a
line with one
endpoint.
A
𝑋
𝑋
𝑌
̅̅̅̅
In symbol: 𝑋𝑌
a) Read as _____.
b) A ____ is a subset of
a line that has two
endpoints.
𝑌
𝑋
𝑍
̅̅̅̅
a) Point 𝑦 divides 𝑋𝑍.
̅̅̅̅ have the
If 𝑋𝑌
same measure with
̅̅̅̅
𝑌𝑍, then ̅̅̅̅
𝑋𝑌 ≅ ̅̅̅̅
𝑌𝑍.
Point 𝑌 is the ____
̅̅̅̅.
of 𝑋𝑍
b) A ____ of a segment
is a point that divides
the segment into two
congruent segments.
Group 4: Given:
𝑌
𝑋
2.
b. Points
W,X,Y,
𝑌
and Z lie
on the
same
plane.
c. Lines 𝑎
and 𝑏
𝑌 𝑍
intersects
at point 𝑐
3.
𝑋
𝑊
𝑋
a) Points𝑊, 𝑋, 𝑌, and 𝑍
lie on the same ___.
b) ___ points are points
that lie on the same
plane.
𝑌
𝑍
6.
𝑎
𝑏
𝐶
8.
Group 5: Given:
𝑊
𝑍
𝑍
5.
𝑛
𝑍
𝑌
𝑌
d. Ray XY
⃗⃗⃗⃗⃗ )
(𝑋𝑌
𝑋 𝑌
7.
a) Points 𝑋, 𝑌 and 𝑍
lie on the same ___.
b) ___ points are points
on the same line.
6
a. Segment
̅̅̅̅)
XY (𝑋𝑌
4.
Group 3: Given:
𝑋
B
1.
𝑋
Group 2: Given:
𝑋
Direction: Match the
physical model/illustration in
column A with the
description/symbol in
column B.
9.
𝑚
e. Line 𝑚 is
parallel to
line 𝑛
𝑚∥𝑛
f. Lines
𝑎, 𝑏, and
𝑐 intersect
at point D.
g. Point Y is
the
midpoint
𝑚
̅̅̅̅.
of 𝑋𝑍
𝑛
h. Lines 𝑚
and 𝑛
are not
coplanar.
i. Line 𝑚 is
perpendic
ular to
line 𝑛
(𝑚 ⊥ 𝑛)
10.
Group 6: Given:
𝑎
𝑏
𝐶
a) Lines 𝑎 and 𝑏
intersect at point __.
b) ___ lines are two
lines with a common
point.
Group 7: Given
𝑛
𝑚
a) Line 𝑚 is
perpendicular to line
𝑛.
In symbol _______.
b) _______ are two
lines intersecting at
right angles.
Group 8: Given:
𝑚
𝑛
a) Line 𝑚 is parallel to
line 𝑛.
In symbol _______.
b) ______ are coplanar
lines that do not
intersect.
Group 9: Given:
𝑏
𝑎
𝑐
𝐷
a) Lines 𝑎, 𝑏, and 𝑐
intersect at point __.
7
j. Points
X,Y,Z are
collinear
points.
k. Points
W,X,Y,Z
are
coplanar
points.
b) ____ are three or
more lines that have
a common point.
Group 10: Given:
𝑘
𝒢
2. Processing the
answer
3. Reinforcing the skills
4. Summarizing the
Lesson
C. Assessment:
b) Lines 𝑚 and 𝑛 are
not coplanar.
Planes 𝒜 and 𝒢
intersect at line ___.
c) _____ lines are two
lines that are not
coplanar.
Questions:
1.
What are the defined terms in geometry?
2.
What are the subsets of a line?
3.
How do we differentiate
a) ray from segment?
b) midpoint from betweenness of point?
c) collinear points from coplanar points?
d) parallel planes from intersecting planes?
4.
How do we describe intersecting lines?
Perpendicular lines? Parallel lines? Concurrent
lines? And skew lines?
5.
What is the symbol for parallel lines? How the
symbol for perpendicular lines?
Use the words parallel, perpendicular, coplanar,
collinear, intersecting, and concurrent to describe how the
figures on the box are related.
L
O
a) Plane 𝒜 and plane ℬ
⃡⃗⃗⃗⃗ and 𝑀𝑇
⃡⃗⃗⃗⃗⃗
E
b) Lines 𝑉𝐸
V
⃡⃗⃗⃗ , 𝑂𝐸
⃡⃗⃗⃗⃗ , and 𝑂𝐻
⃡⃗⃗⃗⃗
c) Lines 𝐿𝑂
H
⃡⃗⃗⃗⃗⃗
⃡⃗⃗⃗⃗⃗
d) Lines 𝑉𝑀 and 𝑀𝑇
M
e) Points V, M, and T
T
A
f) Points M, A, and T
How do you describe and differentiate the defined terms in
geometry such as:
ray and segment; collinear and coplanar points; parallel
and perpendicular lines; intersecting, concurrent and skew
lines?
Describe a mathematical
Use the figure below to
system by changing a word
describe the defined term(s)
in the statement to describe in geometry.
8
the defined terms in
geometry.
1.
2.
1.
𝐴
A segment is a
subset of a line that
has one endpoint/s.
2.
Coplanar points are
points on the same
line.
3.
𝐴
𝐶
𝐴
4.
3.
4.
5.
D. Agreemen:
Noncoplanar lines
that do not intersect
are parallel lines.
Collinear points are
points on the same
plane.
Perpendicular lines
are two lines
intersecting at acute
angles.
Explain briefly:
Is ⃗⃗⃗⃗⃗
𝐴𝐵 = ⃗⃗⃗⃗⃗
𝐵𝐴 ? Why?
̅̅̅̅
Is 𝑃𝑄 = ̅̅̅̅
𝑄𝑃 ? Why?
⃡⃗⃗⃗
⃡⃗⃗⃗
Is 𝑅𝑆 = 𝑆𝑅 ? Why?
V. REMARKS:
VI. REFLECTION:
VII. OTHERS
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
9
𝐵
𝐵
𝑎
𝐵
𝐶
𝑏
5.
𝑎
𝑏
Show the figure of the
following:
a) ⃗⃗⃗⃗⃗
𝐴𝐵 and ⃗⃗⃗⃗⃗
𝐵𝐴
b) ̅̅̅̅
𝑃𝑄 and ̅̅̅̅
𝑄𝑃
⃡⃗⃗⃗
⃡⃗⃗⃗
c) 𝑅𝑆 and 𝑆𝑅
Are they the same? Justify.
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
10
(M8GE-IIIa-1) - Describes a Mathematical System
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
VII. Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 1, Day 3
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry.
The learner is able to formulate an organized plan to
handle a real – life situation.
The learner describes a mathematical system.
M8GE-IIIa-1
Describing a mathematical system (Postulates involving
points, lines, and planes)
Grade 8 Mathematics (Patterns and Practicalities)
pp. 319-320
Geometry (Mathematics Textbook for Third Year High
School)
pp. 14-31
Next Century Mathematics pp. 444-446
Advanced Learners
Average Learners
Perform an activity entitled “REMEMBER ME” (see
separate sheet).
B. Presentation of
the Lesson
1. Problem
Opener/
Group Activity
Activity 2
“WHAT IS MY CONJECTURE?
Direction: Divide the class into five groups. Each group will
be given a task to do within two minutes. After the given
time, output presentation will follow.
ADVANCED LEARNER
AVERAGE LEARNER
A
Group 1: How many lines
Group 1:
can be drawn through a
point?
C
B
11
Group 2: How many lines
can be drawn through two
points?
How many lines can be
drawn through point A only?
Group 2:
A
Group 3: What does the
intersection of two lines look
like?
C
B
Group: 4 What does the
intersection of two planes
look like?
How many lines can be
drawn through points A and
B only?
Group 5: Points G, O, and D
are not on the same line.
How many planes can
contain all three points?
Group 3:
𝑏
C
𝑎
What is the intersection of
line 𝑎 and line 𝑏 ?
Group 4:
G
𝒴
D
O
How many planes can
contain three noncollinear
points?
Group 5:
What is the intersection of
planes 𝒜 and ℬ ?
2. Processing the
answer
Questions:
1.
Based from the activity, what do you think are the
building blocks of geometry?
2.
Can we use these undefined terms to develop other
geometric terms?
3.
What are these defined terms in geometry?
4.
Referring to the statements you have given in the
activity, are they considered to be true?
5.
What do you call a statement which is accepted as
true without proof?
6.
Is postulate important? Why?
7.
Can you cite some postulates involving points, lines
and planes?
12
(The teacher will emphasize the postulates in geometry
specifically involving points, lines, and planes.)
POINT
LINE
PLANE
POINT-EXISTENCE
POSTULATE
STRAIGHT-LINE
POSTULATE
PLANE POSTULATE
s①Space
contains at least
four noncoplanar
points.
②Every line
contains at least
two points.
③Every plane
contains at least
three
noncollinear
points.
Two points
determine a line.
Three noncollinear
points determine a
plane.
LINE-INTERSECTION
POSTULATE
FLAT-PLANE
POSTULATE
If two lines intersect,
then their
intersection is a
point.
If two points are in a
plane, then the line
containing the points
is in the same plane.
PLANE-INTERSECTION
POSTULATE
If two planes
intersect, then their
intersection is a line.
3. Reinforcing the skills
4. Summarizing the
Lesson
C. Assessment:
Suppose “space” consists of only four points A, B, C, and
D, no three of which are collinear,
a) how many planes does “space” contain? Name
them.
b) how many of these planes can contain point A?
point B? point C? point D?
c) how many lines does “space” contain? Name them.
d) how many planes can contain ⃡⃗⃗⃗⃗
𝐴𝐵 ?
What are the postulates involving points, lines, and planes?
Describe a mathematical
system by writing a
description.
a.
𝑛
𝑚
Describe the postulate
involving points, lines, and
planes by completing the
following statements.
1.
Q
2.
b.
ℬ
3.
X
13
Y
If two planes intersect,
then their intersection
is a ______________.
If two __________
intersect, then their
intersection is a point.
Three ________
points determine a
plane.
c.
ℛ
R
4.
Two points determine
a ______.
Y
T
d.
X
ℳ
Y
D. Agreement/
Assignment:
Can you support a notebook on the three fingers?
Must the notebook be supported by all three fingertips?
What postulate is illustrated?
V. REMARKS:
VI. REFLECTION:
VII. OTHERS
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
14
Activity 1
“REMEMBER ME!”
Direction: Distribute the rectangular strips with symbol or term to 10 boys and
another rectangular strips with figure/ illustrations to 10 girls. Let them match the
symbol or term with the correct figure/illustration.
*BOYS*
*GIRLS*
P
1.
̅̅̅̅
𝐴𝐵
2.
⃗⃗⃗⃗⃗
𝐴𝐵
3.
⃡⃗⃗⃗⃗
𝐴𝐵
4.
⃡⃗⃗⃗⃗ ⊥ 𝑃𝑄
⃡⃗⃗⃗⃗
𝐴𝐵
5.
⃡⃗⃗⃗⃗
𝐴𝐵 ∥ ⃡⃗⃗⃗⃗
𝑃𝑄
6.
Intersecting lines
7.
concurrent lines
8.
midpoint
9.
segment bisector
10.
A
B
A
B
Q
A
B
G
B
A
B
A
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝑃𝑄
A
P
15
B
Q
A
B
P
Q
(M8GE-IIIa-1) - Describes a Mathematical System
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 1, Day 4
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry.
The learner is able to formulate an organized plan to
handle a real – life situation.
The learner describes a mathematical system.
M8GE-IIIa-1
Describing mathematical system (Theorems involving
Points, Lines, and Planes)
Geometry (Mathematics Textbook for Third Year High
School)
pp. 42-44
Next Century Mathematics pp. 444-445
Advanced Learners
Average Learners
Activity 1
“CLASSIFY ME!”
Direction: Classify the following statements as ALWAYS
TRUE, SOMETIMES TRUE, or NEVER TRUE.
1. If two lines intersect, then their intersection is a
point.
2. The intersection of two planes is a point.
3. Three noncollinear points determine a plane.
4. Three points that are collinear are also coplanar.
5. Two points determine a plane.
B. Presentation of the
Lesson
16
1. Problem Opener/
Group Activity
Activity 2
“DISCOVER ME!”
Direction: Divide the class into five groups. Each group will
be given a task to do within two minutes. After the given
time, output presentation followed.
Advanced Learners
Average Learners
Group 1:
How many points can be
contained in a line?
Group 1:
Group 2:
How many points can be
contained in a plane?
How many points can be
contained in line 𝑙?
A B C D E F G
𝑙
Group 2:
Group 3:
How many planes can
contain a line and a
point not on the line?
Group 4:
How many planes can
contain two intersecting
line?
Group 5:
How many midpoints
can be contained in a
segment?
𝒜
How many points can be
contained in plane 𝒜?
Group 3:
Y
Z
X
How many planes can
⃡⃗⃗⃗⃗ ) and
contain line XY (𝑋𝑌
point Z?
Group 4:
𝑛
𝑚
How many planes can
contain line 𝑚 and line 𝑛 ?
Group 5:
A
B
How many midpoints can be
contained in segment AB
̅̅̅̅)?
(𝐴𝐵
17
2. Processing the
answer
Questions:
1.
What are the undefined terms mentioned in the
activity?
2.
What are the defined terms mentioned in the
activity?
3.
Based from the statements you have given in the
activity, do you consider them always true?
4.
What do you call a statement that needs to be
proven?
5.
Is theorem important? Why?
6.
Cite some theorems involving points, lines and
planes.
(The teacher will emphasize the theorems in geometry
specifically involving points, lines, and planes.)
1.
2.
3.
4.
5.
3. Reinforcing the skills
A line contains an infinite number of points.
A line and a point not on it lie in exactly one plane.
Exactly one plane contains two intersecting lines.
Every segment has exactly one midpoint.
If a line not contained in a plane intersects the
plane, then the intersection contains only one point.
⃗⃗⃗⃗⃗
6. Given a ray PX and a positive number r. On 𝑃𝑋
|
|
there is one and only point Q, such that 𝑃𝑄 = 𝑟.
Write the theorem that support each statement.
a) In the figure, O is the midpoint of ̅̅̅̅
𝐺𝐷.
G
O
D
b) Line 𝑎 intersects line 𝑏 at point M.
𝑎
𝑏
c) In the figure,
B
R
I
G
H
T
𝑘
line 𝑘 contains many points.
4. Summarizing the
Lesson
C. Assessment:
What are the theorems involving points, lines, and planes?
Write a description for each
illustration applying the
concept of theorems
involving points, lines, and
planes.
Describe the theorems
involving points, lines, and
planes by completing the
following statements.
1.
1.
I L O V E M A T H
18
A line contains an
____ number of
points.
2. A line and a point not
on it lie in exactly one
________.
𝑥
2.
3. Exactly one plane
contains _______
intersecting lines.
4. Every _______ has
exactly one midpoint.
5. If a line not contained
in a plane intersects
the plane, then the
intersection contains
only _____ point.
𝒜
3.
A
B
C
𝑎
4.
𝑏
5.
D. Agreement/
Assignment:
A
R
T
1. Given a line with a coordinate system, is the point
with coordinate O the midpoint of the line?
2. Does a line have a midpoint? Why?
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
19
(M8GE-IIIa-c-1) - Illustrates the need for an axiomatic structure
of a mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
School
Grade Level
Learning
Area
Teacher
Time &
Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Quarter
8
MATHEMATICS
THIRD
Week 2, Day 1
The learner demonstrates understanding of key
concepts of axiomatic structure of geometry and triangle
congruence.
The learner is able to formulate an organized plan to
handle a real-life situation.
Illustrates the need for an axiomatic structure of a
mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
M8GE – IIIa-c- 1
II.
III.
CONTENT
Illustrating undefined terms
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s
Materials pages
3. Textbook pages
4. Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning EXPLORE – Worktext in Mathematics 3 by
Resources
Nellie Toliao-Dasco ; pp. 1-3
Geometry – Textbook for Third Year by Cecile M. De
Leon, Soledad Jose-Dilao and Julieta G.
Bernabe ; pp. 2- 10
IV.
PROCEDURE
Advanced
Average Learners
Learners
A. Preliminary Activities/ The picture below shows representations of geometric
Motivation
figures abound in nature. Can you find representations
of a point? of a line? of a plane?
20
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
(Work in pairs.)
2. Processing the
Answer
3. Reinforcing the
Skills
4.Summarizing the
Lesson
C. Assessment
D. Agreement/
Assignment
Instructions:
Instructions:
1. Draw a rectangular top 1. Draw a rectangular
of a table.
top of a table.
2. Mark each corner by 2. Mark each corner by
capital letters A, B, C
capital letters A, B, C
and D.
and D.
Questions:
Consider the figure below
1. Which part of the table to complete the table.
represent
A
B
points?_____________
Name
the
points.
______
2. Which parts of the table
represent lines? ______
D
C
Name the lines. ______
POINTS
LINES
PLANE
3. Which part of the table
represent a plane?____
Which
Name the plane.______
parts
of the
table
represents;
Name
the;
1. What did you consider in completing and answering
correctly the activity?
2. How will you describe points, lines and planes?
3. How do we classify these terms: points, lines and
planes?
Illustrate each of the following and label the diagram.
(By group/by pair)
1. Point X lies on plane Y.
2. Points R, S, T and U lie on line 𝑛.
3. Plane B contains CD.
4. GH intersects plane 𝒜 at point E.
5. Lines 𝑎 and 𝑏 intersect at C.
How do you illustrate points, lines, and planes?
Illustrate the following and label each diagram.
1. Point B lies in plane ℳ.
2. Lines 𝑙 and 𝑚 intersect at point T.
3. Line EF is on plane 𝒢.
4. Planes 𝒜 and ℬ intersect at line PR.
Given:
E
A
C
B
21
D
Identify the following:
1. Points on plane R
2. Points on line AD
3. Points on line BE
4. Lines on the plane R
5. Plane
V.
VI.
VII.
A.
B.
C.
D.
E.
F.
G.
REMARKS
REFLECTION
OTHERS
No. of learners
who earned 80%
on the formative
assessment
No. of learners
who
require
additional activities
for remediation.
Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
No. of learners
who continue to
require
remediation.
Which
of
my
teaching strategies
worked well? Why
did it work?
What
difficulties
did I encounter
which my principal
or supervisor can
help me solve?
What innovation or
localized
material/s did I
use/discover
which I wish to
share with other
teachers?
22
(M8GE-IIIa-c-1) - Illustrates the need for an axiomatic structure
of a mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
School
Grade Level
Learning
Area
Teacher
Time &
Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Quarter
8
MATHEMATICS
THIRD
Week 2, Day 2
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to formulate an organized plan to
handle a real-life situation.
Illustrates the need for an axiomatic structure of a
mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
M8GE – IIIa-c- 1
II.
III.
CONTENT
Illustrating undefined terms
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s
Materials pages
3. Textbook pages
4. Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning EXPLORE – Worktext in Mathematics 3 by
Resources
Nellie Toliao-Dasco ; pp. 1-3
Geometry – Textbook for Third Year by Cecile M. De Leon,
Soledad Jose-Dilao and Julieta G. Bernabe ;
pp. 2- 10
IV.
PROCEDURE
Advanced Learners
Average Learners
A. Preliminary Activities/ Tell whether each of the following suggests a point, a line
Motivation
or a plane.
1. tip of a box
4. cover of a book
2. a corner of a room
5. tip of a pen
3. star in the sky
6. A taut clothesline
23
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
(Work in pairs.)
Match column A with column B by describing the illustrated
figures.
COLUMN A
g
1.
f
a. lines AD and BC lie
on plane ℰ
A
b. plane 𝒩 and plane ℒ
intersect at BC
B
2.
COLUMN B
A
𝒩
3.
ℰ
C
B
c. line 𝑔 and line 𝑓
intersect at point A
D
A
4.
d. points C, D and E lie
on the same plane
D
ℒ
C
e. plane 𝒩 and line
AB intersect at point A
5.
2. Processing the
Answer
 How do you describe the figures?
 What should be considered to describe the figures
correctly?
3. Reinforcing the
Skills
Illustrate the following and label each diagram correctly.
1. line A is ⊥ to line B
1. line A is ∥ to line B
2. points L, O, V and E lie
2. points F,A,I and T lie on
on plane ℛ
plane ℋ
3. line MI and line NE
3. line YO and line UR
intersects at point U
intersect at point S
4. planes 𝒞 and 𝒜 have
4. planes 𝒞 and 𝒜
RE in common
intersect at RE
5. line EL and plane ℐ
5. line EL and plane 𝒵
intersect at point L
intersect at point L
How do you illustrate points, lines, and planes?
4. Summarizing the
Lesson
24
C. Assessment
D. Agreement/
Assignment
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
Illustrate the following and label each diagram correctly.
1. points A and B lie on the
1. planes 𝒜 and ℬ
same line
intersect at DC
2. points A and B are
2. planes ℰ and 𝒟 have AB
collinear
in common
3. plane ℬ and line AC met 3. line AB is parallel to line
CD
at point C
4. lines AC and DE
4. points A,B,C and D lie
intersect at point B
are coplanar
5. points A, B, C and D lie
5. line AC and line DE
on the same plane
intersect at point B
List down things that illustrate points, lines and planes.
Give 5 examples for each.
25
(M8GE-IIIa-c-1) - Illustrates the need for an axiomatic structure of
a mathematical system in general and in Geometry in particular
DEFINED TERMS and UNDEFINED TERMS
School
Teacher
Time and
Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Grade Level
Learning Area
Quarter
8
Mathematics
Third
Week 2, Day 3
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to formulate an organized plan to
handle a real-life situation.
Illustrates the need for an axiomatic structure of a
mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
M8GE-IIIa-c-1
II.
III.
CONTENT
Illustrating defined terms
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional
Materials
from
Learning Resource
(LR) portal
B. Other
Learning EXPLORE – Worktext in Mathematics 3 by
Resources
Nellie Toliao-Dasco ; pp. 1-3
Geometry – Textbook for Third Year by Cecile M. De Leon,
Soledad Jose-Dilao and Julieta G. Bernabe ; pp. 2- 10
IV.
PROCEDURE
Advance Learners
Average Learners
A. Preliminary Activities/ Name the figures below.
Match column A with
Motivation
1.
column B and try to
describe each.
A
B
COLUMN A
1.
A
2.
A
2.
3.
3.
26
B
B
COLUMN B
1. angle CAB
2. line segment BA
3. triangle ABC
4. ray AB
5. line AB
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
Instruction: (By group/by pair)
1. Draw a rectangular top of a table.
2. Mark each corner by capital letters E, L, S and A.
3. Extend the upper side and the lower side, the right and
left sides and put arrows each end.
(The figure can be presented to the average group.)
2. Processing the
Answer
3. Reinforcing the
Skills
4.Summarizing the
Lesson
C. Assessment
A
B
C
D
1. Name the extended sides of the table.
2. Name the following parts by the extended sides;
a. line segments
b. rays
c. angles
(The teacher will discuss further the parts of the figures
illustrated.)
 What figures were formed by the extended
sides?
 How do you describe each figure?
 What do you call each?
Illustrate each of the following figures and label the
diagram.
1. segment AB
2. ray XY
3. angle ABC
4. point B is the midpoint of segment AC
5. point P is in the exterior of ∠ 𝐴𝐵𝐶
6. point Q is in the interior of ∠ 𝐷𝐸𝐹
7. ∠𝐴𝐵𝐶 is an acute angle
8. ∠𝑀𝑁𝑂 is an obtuse angle
9. ∠𝑋𝑄𝑃 is a right angle
10. ray OP bisect ∠ 𝐴𝑂𝐶
Illustrate the need for axiomatic structure in the
illustration of a ray, segment, angle?
Illustrate each of the following figures and label the
diagram.
1. point Z divides
1. point Z is the endpoint
segment XY
of segment XY
congruently
2. EF bisects ∠𝐶𝐸𝐷
27
D. Agreement/
Assignment
3. ray AB
2. ray EF divides ∠ 𝐶𝐸𝐷
4. ∠ 𝑂 is an obtuse angle
into two congruent
adjacent angles
point N is the vertex of
3. ray BA is on line AC
∠ 𝑀𝑁𝑂
4. ∠ 𝑃 is a right angle
point N is the common
endpoint of the of
∠ 𝑀𝑁𝑂
Use the figure to complete the table below.
A
G
B
F
C
Number of figures
formed
Angles
Rays
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s did
I use/discover which I
wish to share with
other teachers?
28
Line
Segments
E
D
Name of the figure
Angles
Rays
Line
Segments
(M8GE-IIIa-c-1) - Illustrates the need for an axiomatic structure
of a mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
School
Teacher
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
III.
A.
1.
2.
3.
4.
B.
IV.
Grade Level
Learning Area
Quarter
8
Mathematics
Third
Week 2, Day 4
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to formulate an organized plan to
handle a real-life situation.
Illustrates the need for an axiomatic structure of a
mathematical system in general and in Geometry in
particular DEFINED TERMS and UNDEFINED TERMS
M8GE – IIIa-c- 1
CONTENT
Illustrating defined terms (Triangles)
LEARNING
RESOURCES
References
Teacher’s Guide
pages
Learner’s
Materials pages
Textbook pages
Additional
Materials
from
Learning
Resource
(LR)
portal
Other
Learning EXPLORE – Worktext in Mathematics 3 by
Resources
Nellie Toliao-Dasco ; pp. 1-3
Geometry – Textbook for Third Year by Cecile M. De Leon,
Soledad Jose-Dilao and Julieta G. Bernabe ;
pp. 2- 10
PROCEDURE
Advance Learners
Average Learners
A. Preliminary Activities/
Motivation
Plot 3 non-collinear points
and connect the points
consecutively. What figure
is formed?
29
Consider the following
pictures. What figures do
they have in common?
1. Study each group of triangles below to answer the
questions that follow.
GROUP A
GROUP B
A
A
10 cm
B. Presentation of the
Lesson
1. Problem Opener
B
90°
B
C
10 cm
C
B
B
60°
A
60°
60°
30
C
A
9 cm
C
A
A
120°
C
B
B
7 cm
C
a. How do you classify triangles?
b. How does each triangle in each group differ from
each other?
c. What do we call each?
2. Given: ΔABC
A
F
D
B
E
C
Name the primary and secondary parts of ΔABC.
Primary Parts:
a. Vertex
b. Angles
c. Sides
Secondary Parts:
a. Median
b. Angle Bisector
c. Altitude
2. Processing the
Answer
3. Reinforcing the
Skills
4.Summarizing the
Lesson
C. Assessment
 What is a triangle?
 What are the parts of a triangle?
 How do we illustrate a triangle?
Illustrate the following and label the diagram.
1. points A,B, and C are
1. AC, CB and BA are the
the vertices of ΔABC
sides of ΔABC
2. ray BD is an angle
2. AE is a median of
bisector of ΔABC
ΔABC
3. ΔABC has equal sides
3. ΔABC is a right triangle
4. ∠𝐴 is an obtuse angle of 4. ΔABC has no equal
sides
ΔABC
How do you illustrate a triangle and its parts?
Illustrate the triangle given the following parts.
1. vertices: F, U and N
4. median: FY
2. angles: F, U and N
5. angle bisector: UX
3. sides: FU, UN and NF
6. altitude: ZN
31
D. Agreement/
Assignment
Illustrate the parts of the triangle below.
1. sides: XY, YZ and ZX
2. median: BX
3. angle bisector: AZ
4. altitude: CY
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which
of
my
teaching strategies
worked well? Why
did it work?
F. What difficulties did I
encounter which my
principal
or
supervisor can help
me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
32
(M8GE-IIIa-c-1) - Illustrates the need for an Axiomatic Structure
of a Mathematical System in Geometry (Postulates)
GRADE
LEVEL
LEARNING
AREA
SCHOOL
TEACHER
DATE OF
TEACHING
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies
/
LC Code
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Material Pages
3. Textbook
Pages
4. Additional
Materials from
Learning Resource
(LR) Portal
B. Other Learning
Resource
IV. PROCEDURE
A. Motivation
QUARTER
8
Mathematics
THIRD
Week 3, Day 1
The learner demonstrates understanding of key concepts of
axiomatic structure of geometry and triangle congruence.
The learner is able to formulate an organized plan to handle a reallife situation.
The learner illustrates the Need for an axiomatic structure of a
mathematical system in general, and in Geometry in particular: (a)
defined terms; (b) undefined terms; (c) postulates; and (d)
theorems
LC CODE: M8GE-IIIa-c-1
Illustrating the need for an Axiomatic Structure of a Mathematical
System in Geometry (postulates)
*De Leon, Dilao, Bernabe, “Geometry (Textbook)”, JTW
Corporation, 2009 pp. 3-4
*Institute for Science and Mathematics Education Development,
“Geometry III (Textbook)”, Capitol Publishing House, Inc. Copyright
1978 and 1988, pages 258 – 259
Advanced Learner
Average Learner
DIRECTIONS.
Rearrange the letters in each word to complete the sentence.
1. A tniop sha on niosnemid.
2. A neli sha ylno htgnel.
Rearrange the phrases to form a complete statement.
“without proof which the validity a postulate is or truth is
assumed a statement of”
33
B. Presentation
of the Lesson
1. Opener/
Activity
Advanced Learner
Average Learner
How do you identify whether a mathematical statement is a
postulate or not?
ACTIVITY 1
INVESTIGATE ME!
On this set of activities, we are going to investigate more on the
details of postulates on points and lines.
1. Plot two points on a ¼ sheet of paper. Name the points A and B.
2. Connect the two points using a line. Name the line 1.
3. Plot another point. Make sure that the point does not lie on line 1
(non-collinear). Name the point C.
4. Connect both point A and B to point C. Name the line 2 and 3.
QUESTIONS:
a. How many lines have you made?
b. Atleast how many points do you need for you to make a line?
c. Is it possible for a line to contain three or more non-collinear
points?
Activity # 2
1. Plot a point on a ¼ sheet of
paper. Name the point O.
2. Plot another point on the right
side of point O. Name it K.
3. Measure the distance
between points O and K.
4. Plot another point on the left
of point O and name it S.
5. Measure the distance
between points O and S.
6. Plot another point at the
bottom of O and name it H.
7. Measure the distance
between points O and H.
QUESTIONS:
a. What are the distances?
b. Is it possible to have a
negative value of distance even
if you change the direction of a
point?
2. Processing
the Answer
Processing Questions:
1. Were you able to follow the procedures correctly and answer
all the questions?
2. What observations can you make out of the activities (activity
# 1 and # 2)
3. Based from your observations, what conclusion can you give
for each activity?
4. Will your conclusion be true to all other examples? Why or why
not?
5. How will you illustrate other examples?
34
3. Reinforcing
the Skills
“ILLUSTRATE ME”
DIRECTION. Follow the directions in illustrating the postulate below.
“The points of a line and the set of numbers can be put into a oneto-one correspondence in such a way that if “a” is the number
associated with point A, and “b” is the number associated with point
B, then the distance |𝐴𝐵| is |𝑎 − 𝑏|.”
a. Sketch a number line from -5 to 5.
b. Label -5 as point A, -4 as point B, -3 as point C, -2 as point
D, -1 as point E, 0 as point F, 1 as point G, 2 as point H, 3
as point I, 4 as point J and 5 as point K.
c. Tell the distance of the following:
1. point A to point B
2. point C to point G
3. point B to point K
d. If you get the distance of two points will it have negative
measures?
4. Summarizing
the Lesson
C. Assessment
D. Assignment
1. How do you illustrate postulates on points and lines?
Illustrate the given postulate by elaborating, giving examples or
sketch.
Given two points A and B of a line, a coordinate system can be
chosen in such a way that the coordinate of A is zero and the
coordinate of B is positive.
Illustrate the following postulates:
1. Given a line, there is a point not on the line
2. Given a plane, there is a point not on the plane.
Remarks:
A. Number
of
Learners who
earned 80% in
the formative
assessment.
B. Number
of
Learners who
require
additional
activities
for
remediation.
C. Number
of
Learners who
caught up with
the lesson.
D. Number of
learners who
continue to
require
remediation.
35
(M8GE-IIIa-c-1) - Illustrates the need for an Axiomatic Structure
of a Mathematical System in Geometry (Postulates)
SCHOOL
GRADE LEVEL
LEARNING
AREA
TEACHER
DATE OF
TEACHING
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
LC Code
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Material Pages
3. Textbook
Pages
4. Additional
Materials from
Learning Resource
(LR) Portal
B. Other Learning
Resource
IV. PROCEDURE
A. Motivation
QUARTER
8
Mathematics
THIRD
Week 3, Day 2
The learner demonstrates understanding of key concepts of
axiomatic structure of geometry and triangle congruence.
The learner is able to formulate an organized plan to handle a
real-life situation.
The learner illustrates the need for an axiomatic structure of a
mathematical system in Geometry (postulates)
LC CODE: M8GE-IIIa-c-1
Illustrating the need for an Axiomatic Structure of a
Mathematical System in Geometry (Postulates)
*De Leon, Dilao, Bernabe, “Geometry (Textbook)”, JTW
Corporation, 2009 pp. 3-4
*Institute for Science and Mathematics Education
Development, “Geometry III (Textbook)”, Capitol Publishing
House, Inc. Copyright 1978 and 1988, pages 258 – 259
Advanced Learner
Average Learner
FORMING PLANES and ANGLES
DIRECTION. List down or sketch the different plane figures
that you can see or make out of the figure below.
How many plane figures can you see or sketch?
What kinds of angles can you see?
36
B. Presentation
of the Lesson
1. Opener/ Activity
Advanced Learner
Average Learner
What is a plane or plane figure? How about an angle?
How do you draw plane figures? How do you draw angles?
ACTIVITY 1
EXPLORE ME!
On this set of activities, we are going to explore more on the details
of postulates on planes.
1. Given triangle ABC.
A
2.
3.
4.
5.
B
C
Plot a point anywhere between points A and B. Name it D.
Plot another point (not the same with D) between A and B.
Name it point E.
Plot a point anywhere between points A and C and another
point anywhere between points B and C. Name the points
F and G, respectively.
Connect the points to form lines.
QUESTIONS:
a. How many lines have you formed?
b. Are the figures still on the plane?
c. If you use two points anywhere on the plane to make a line,
will the newly formed line still lie on the plane? Why or why
not?
Activity # 2
Draw a straight line and name it line M.
Plot two points on line M and name it point K and S.
Plot a point not on line M and name it point O.
Connect points K and O using a ray. Measure the angle
formed by points O, K and S.
5. Plot another point not on line M and name it R. Connect
points K and R using a ray. Measure the angle formed by
points R, K and S.
1.
2.
3.
4.
QUESTIONS:
a. What are the measures of the different angles?
b. Are they equal or not?
c. If you are going to repeat step 5 for several times, will you
still produce different measures of angles? Why or why
not?
37
2. Processing the
Answer
Processing Questions:
1. Were you able to follow the procedures correctly and
answer all the questions?
2. What observations can you make out of the activities
(activity # 1 and # 2)
3. Based from your observations, what conclusion can you
give for each activity?
4. Will your conclusion be true to all other examples? Why or
why not?
5. How will you illustrate other examples?
3. Reinforcing the
Skills
SHOW ME MORE
On each column is a set of postulate and a procedure to
illustrate it. Rearrange the procedure properly by numbering
each from 1- 4.
“If two distinct planes intersect, “If B, A and C are collinear
then their intersection is a points and D is not a point on
line.”
line BC, then 𝑚∠𝐵𝐴𝐷 +
𝑚∠𝐷𝐴𝐶 = 180.”
Sketch another plane that
Plot another point D which
intersects
the
former
is not on line BC.
plane.
Choose another two plane
and make them intersect.
Highlight the intersection.
Highlight the part where
the two planes intersect.
Sketch a plane (triangle,
square, rectangle) in a
paper. Choose from the
given in the parenthesis.
Measure angle BAD and
DAC. Get the sum of
𝑚∠𝐵𝐴𝐷 + 𝑚∠𝐷𝐴𝐶.
Draw a line to connect
points A and D.
Plot three collinear points
on a paper. Label them
points B, A and C,
respectively. Connect the
points to make line BC.
4. Summarizing
the Lesson
How do you illustrate postulates on planes and angles?
C. Assessment
DIRECTION. Illustrate the following postulate by elaborating,
giving examples or sketch.
D. Assignment
1. Any three points lie in at least one plane and any three noncollinear points lie in exactly one plane.
2. If B, A and C are not collinear and D is in the interior of
∠𝐵𝐴𝐶 then 𝑚∠𝐵𝐴𝐷 + 𝑚∠𝐷𝐴𝐶 = 𝑚∠𝐵𝐴𝐶.
Research on the other postulates on planes and angles.
38
Remarks:
A. Number of
Learners who
earned 80% in the
formative
assessment.
B. Number of
Learners
who
require additional
activities
for
remediation.
C. Number of
Learners
who
caught up with the
lesson.
D. Number of
learners who
continue to
require
remediation.
39
(M8GE-IIIa-c-1) Illustrates the Need for an Axiomatic
Structure of a Mathematical System in Geometry (Theorems)
SCHOOL
GRADE LEVEL 8
LEARNING
Mathematics
AREA
THIRD
QUARTER
Week 3, Day 3
TEACHER
DATE OF
TEACHING
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies
/
LC Code
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Material Pages
3. Textbook
Pages
4. Additional
Materials from
Learning
Resource (LR)
Portal
B. Other Learning
Resource
IV. PROCEDURE
A. Motivation
The learner demonstrates understanding of key concepts of
axiomatic structure of geometry and triangle congruence.
The learner is able to formulate an organized plan to handle a reallife situation.
The learner illustrates the need for an axiomatic structure of a
mathematical system in general, and in Geometry in particular: (a)
defined terms; (b) undefined terms; (c) postulates; and (d)
theorems
LC CODE: M8GE-IIIa-c-1
Illustrating the need for an Axiomatic Structure of a Mathematical
System in Geometry (theorems)
*De Leon, Dilao, Bernabe, “Geometry (Textbook)”, JTW
Corporation, 2009 pp. 3-4
*Institute for Science and Mathematics Education Development,
“Geometry III (Textbook)”, Capitol Publishing House, Inc.
Copyright 1978 and 1988, pages 259 – 262
Advanced Learner
Average Learner
“What’s the Truth?”
DIRECTION. Tell whether the given statement is true or false.
1. A theorem is a statement that does not need to be
proven.
2. A theorem is made using undefined terms, defined terms,
and postulates.
3. Geometry is derived from two Greek words geo meaning
earth and metrein meaning time.
40
B. Presentation
of the Lesson
1. Opener/ Activity
Advanced Learner
Average Learner
What are the similarities and differences between a postulate and
a theorem?
What are the different postulates discussed previously?
PROVE ME!
On this set of activities, we are going to make a proof on the
different theorems.
Theorem # 1
“A line contains an infinite number of points.”
Arrange the following statements to form the complete proof.
1. The points on a line and the set of numbers can be put into
one-to-one correspondence.
2. Therefore, by the definition of one-to-one correspondence,
the points on a line is an infinite set.
3. The set of numbers is infinite.
Theorem # 2
“If in a triangle two sides are not congruent, then the angles
opposite these sides are not congruent and the angle opposite the
longer side is the larger angle.”
Arrange the following steps in order to make a complete proof and
answer the questions below.
1. Make a scalene triangle.
2. Measure the length of the sides.
3. Measure the angles.
2. Processing
the Answer
3. Reinforcing
the Skills
QUESTIONS:
a. Compare the sides and their opposite angles. Are there
sides with the same measure of the angles?
b. Identify the shortest side. Is it opposite the smallest angle?
c. How about the longest side, is it opposite the largest
angle?
1. Were you able to answer the questions correctly?
2. Are your answers true to all other examples? Why or why not?
3. How are you able to illustrate the theorem?
Illustrate the theorem by making a complete proof.
“If two coplanar lines are perpendicular to the same line, then the
two coplanar lines are parallel.”
4.Summarizing
the Lesson
How do you illustrate theorems on points and lines?
C. Assessment
Illustrate the theorem. You can use a sketch or give examples in
your proof.
Theorem: “The sum of the lengths of two sides of a triangle is
greater than the length of the third side.”
41
D. Assignment
Illustrate the proof of the theorem.
In a triangle, the line segment joining the midpoints of two sides
is parallel to the third side, and its length is one-half the length of
the third side.
Remarks:
A. Number of
Learners
who
earned 80% in the
formative
assessment.
B. Number of
Learners
who
require additional
activities
for
remediation.
C. Number of
Learners
who
caught up with the
lesson.
D. Number of
learners who
continue to
require
remediation.
42
(M8GE-IIIa-c-1) - Illustrates the Need for an Axiomatic
Structure of a Mathematical System in Geometry (Theorems)
SCHOOL
GRADE LEVEL
LEARNING
AREA
TEACHER
DATE OF
TEACHING
I. OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies
/
LC Code
II. CONTENT
III. LEARNING
RESOURCES
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Material Pages
3. Textbook
Pages
4. Additional
Materials from
Learning
Resource (LR)
Portal
B. Other
Learning
Resource
IV. PROCEDURE
A. Preliminary
Activity
QUARTER
8
Mathematics
THIRD
Week 3, Day 4
The learner demonstrates understanding of key concepts of
axiomatic structure of geometry and triangle congruence.
The learner is able to formulate an organized plan to handle a reallife situation.
The learner illustrates the need for an axiomatic structure of a
mathematical system in general, and in Geometry in particular: (a)
defined terms; (b) undefined terms; (c) postulates; and (d)
theorems
LC CODE: M8GE-IIIa-c-1
Illustrating the need for an Axiomatic Structure of a
Mathematical System in Geometry (Theorems)
*De Leon, Dilao, Bernabe, “Geometry (Textbook)”, JTW
Corporation, 2009 pp. 3-4
*Institute for Science and Mathematics Education Development,
“Geometry III (Textbook)”, Capitol Publishing House, Inc.
Copyright 1978 and 1988, pages 259 – 262
Advanced Learner
Average Learner
“CONCEPT TILES”
DIRECTION. Create the correct sentence by moving the tiles
which contain a word/phrases.
A
needs
proven
that
statement
to be
is a
theorem
a true
43
A
Theorem
is a
true
statement
that
needs
to be
proven.
ANSWER:
B. Presentation
of the Lesson
1. Opener/ Activity
Advanced Learner
Average Learner
What are the different undefined terms? The different defined
terms?
What is a postulate? What is a theorem?
THE AXIOMATIC BINGO!
Goal: To conduct a bingo game in class where the concepts on
undefined terms, defined terms, postulates and theorems are
used.
Role: Bingo Player
Audience: Students
Situation: The teacher will present bingo cards to his/her
students. Each card is composed of a 3x3 square table. Each card
is unique and different from the others. Each number on the card
has a question to be answered by the students. At the start, the
teacher will give a pattern (triangle, square, blackout) to be
followed by each bingo player. The student/bingo player who can
complete the pattern first will be declared as the winner.
Performance: All the students are involved because each has a
bingo card.
Standard: The number of correct answers to the given questions
will determine his/her score in the performance task.
BINGO CARD Appearance: (the numbers on the card will vary
for each card)
AXIOMATIC BINGO CARD
Goodluck and Enjoy! No. ______
1
2
3
4
5
6
7
8
9
44
Remarks:
A. Number of
Learners
who
earned 80% in the
formative
assessment.
B. Number of
Learners
who
require additional
activities
for
remediation.
C. Number of
Learners
who
caught up with the
lesson.
D. Number of
learners who
continue to
require
remediation.
45
(M8GE-IIId-1) - Illustrates Triangle Congruence
School
Teacher
Grade Level
Learning
Area
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
LC Code
II.
CONTENT
Quarter
8
MATHEMATICS
THIRD
Week 4, Day 1
The learner demonstrates understanding of the key
concepts of triangle congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representation.
The learner illustrates triangle congruence
M8GE-IIId-1
Illustrating triangle congruence using the basic parts of a
triangle.
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s
pp.349 – 352
Materials pages
3. Textbook pages
4. Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning
Resources
IV.
PROCEDURE
Advance Learners
Average Learners
A. Preliminary Activities/ The teacher shows to the class the location of the
Motivation
Bermuda Triangle and may give trivia about it.
B. Presentation of the
What geometric figure is talked about in the trivia?
Lesson
Are you still familiar with triangles?
1. Problem Opener/
Activity
How important is it to talk about triangles and its
properties?
“KNOW ME MORE, IT’S BETTER!”
(Group size may be determined by the teacher)
The teacher distributes two cut-outs of triangles to every
group. (See attached Activity Card)
46
Group 1: Name the given triangles.
List the primary parts of each triangle.
 Vertices
 Sides
 Angles
Group 2: Name the given triangles.
List the secondary parts of each triangle.
 Median
 Altitude
Group 3: Describe and illustrate (2 triangles)
 Included side
 Included angle
Group 4: Describe and illustrate (2 triangles)
 Opposite side
 Opposite angle
 Adjacent sides
Group 5: Illustrate and identify the parts of
 Isosceles triangle
 Right triangle
1. Processing the
Answer








2. Reinforcing the
Skills
4.
Summarizing the
Lesson
What is a triangle?
How many vertices, sides, and angles are there in
a triangle?
What can you say about the median and the
altitude of the triangle?
How will you describe the included side and the
included angle in a triangle?
When do we say that a side is opposite an angle?
When do we say that a side is adjacent to each
other?
How do we differentiate an isosceles triangle from
a right triangle?
Is it possible to have two right angles in a triangle?
Why or why not? How about two obtuse angles?
How important is it to study triangles?
Accomplish the profile of a Triangle (Use the separate
worksheet)
What is a triangle?
What are the parts of a triangle?
What are the other concepts related to triangles?
47
C. Assessment
A. Identify whether each statement is true or false. If
TRUE draw smiley if FALSE draw a sad face.
1. A triangle has 3 sides and 3 angles.
2. The altitude of a triangle is the line segment from
one vertex perpendicular to the opposite side.
3. A vertex of a triangle is the center of a triangle.
4. The median is the line segment connecting the
midpoints of any two sides.
5. An include angle is the angle formed by the two
adjacent sides.
D. Agreement/
Assignment
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s did
I use/discover which I
wish to share with
other teachers?
48
ACTIVITY CARD FOR BASIC CONCEPTS INVOLVING TRIANGLE
GROUP 1: Name each triangle and list down the primary parts.
S
M
R
D
I
E
GROUP 2: Name each triangle and list down the secondary parts.
P G
L
C
R
49
U
GROUP 3:
Describe and illustrate (2 triangles)
 Included side
 Included angle
G
W
F
L
I
GROUP 4: Describe and illustrate (2 triangles)
 Opposite side
 Opposite angle
 Adjacent sides
 Adjacent angles
G
W
H
Y
S
50
P
GROUP 5: Illustrate and identify the parts of
 AN Isosceles triangle
 Right triangle
X
Z
Y
S
U
T
51
(M8GE-IIId-1) - Illustrates Triangle Congruence
School
Teacher
Grade Level
Learning
Area
Time & Date
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
Quarter
8
MATHEMATICS
THIRD
Week 4, Day 2
The learner demonstrates understanding of the key
concepts of triangle congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving congruent
triangles using appropriate and accurate representation.
The learner should be able to illustrate triangle congruence.
C. Learning
Competencies/
M8GE-IIId-1
LC Code
II.
CONTENT
Illustrating congruent triangles
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
pp.349 - 352
Materials
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning
Resources
IV.
PROCEDURE
Advanced Learners
Average Learners
A. Preliminary
Activities/
“King’s Wish. Bring Me.”
Motivation
 Pair of earrings
 Pair of shoes
 Eye glasses
 Pair of notebooks
B. Presentation of the
Lesson
1. Problem Opener
What did you notice about these pairs of objects?
Describe their shapes and sizes.
Have you seen the bridges? How about the towers? What
shapes of braces are used to reinforced the stability of these
structures? Why do you think they were used?
52
“To whom Should I Pair?”
2. Group Activity
(Group size may be determined by the teacher)
(The teacher distributes cut-outs of triangles to every group.)
At a count of five, find your partner holding the same shape
of triangle as yours and shout “BASIC LANG!” if you are done.
You will be given a prize. Just stay in front/ beside your
partner for some questions/ instructions.





Name your triangles.
What kind of triangles are they?
Why did you say that they are congruent? What is the
symbol for congruence?
Which parts of one triangle corresponds to the other?
What is the symbol for correspondence?
Each group will be given a pair of congruent triangles and
they will write the corresponding parts of the triangles.
Group 1: pair 1
Group 2: pair 2
Group 3: pair 3
Group 4: pair 4
Group 5: pair 5
3. Processing the
Answer






4. Reinforcing the
Skills
What are congruent triangles?
What is the symbol for congruence?
How many congruent corresponding parts are there?
What is the symbol for correspondence?
How do you put identical markings on congruent
corresponding parts? Illustrate.
Where can you see/find congruent triangles?
Exercises.
Refer to the figure.
53
1.
ABD ≅ CBD. Write down the six pairs of congruent
corresponding parts.
2. Draw ∆𝑀𝐴𝑋 and ∆𝐾𝐹𝐶 where
MA ≅ KF,
AX ≅ FC,
MX ≅ KC,
M ≅ K
 A ≅ F
 X ≅ C
3. Which of the following shows the correct congruence
statement for the figure below?
a. ∆𝑃𝑄𝑅 ≅ ∆𝐾𝐽𝐿
b. ∆𝑃𝑄𝑅 ≅ ∆𝐿𝐽𝐾
c. ∆𝑃𝑄𝑅 ≅ ∆𝐿𝐾𝐽
d. ∆𝑃𝑄𝑅 ≅ ∆𝐽𝐿𝐾
5.Summarizing
When are two triangles congruent?
the
Lesson
C. Assessment
Exercises. Illustrate the triangle congruence by list down and
putting identical markings on the congruent corresponding
parts.
1. Write the six congruent corresponding parts. Refer to
the figure below.
2. Draw the figure and mark the identical congruent
corresponding parts of
∆𝑅𝑈𝐵 ≅ ∆ 𝑇𝑈𝐺.
D. Agreement/
Assignment
V.
REMARKS
VI.
REFLECTION
Draw two congruent triangles and identify their congruent
corresponding parts.
54
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners who
continue to require
remediation.
E. Which
of
my
teaching strategies
worked well? Why
did it work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized material/s did
I use/discover which I
wish to share with other
teachers?
55
(M8GE-IIId-1) - Illustrates Triangle Congruence
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
Grade Level
Learning Area
Quarter
8
MATHEMATICS
THIRD
Week 4, Day 3
The learner demonstrates understanding of the key
concepts of triangle congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representation.
The learner illustrates triangle congruence.
M8GE-IIId-1
Illustrating congruent triangles
380-392
292-294
Moving Ahead with Mathematics II, 1999 (pages 114-115)
Advanced Learners
Average Learners
“FIND YOUR PARTNER”
Your group will be given five pairs of congruent figures, each shape for each
member. At the count of three, find your partner by matching the shape that you
have with another’s shape.
1. Why/How did you choose your partner?
2. How will you describe the two figures you have?
3. What can you say about the size and shape of the
two figures?
4. When do we have congruent triangles?
5. How many pairs of corresponding parts are
congruent if two triangles are congruent?
B. Presentation of the
Lesson
1. Problem Opener:
2. Group Activity
What must be the corresponding parts that are congruent
so that the two triangles are congruent?
The class will be divided into 4 groups to perform an
activity. Discuss the answer to the class.
56
Group 1
List the six pairs of corresponding congruent parts in the
given congruent triangles.
Given: ∆ 𝐴𝑅𝑇 ≅ ∆𝐴𝑅𝑀
Group 2
List the six pairs of corresponding congruent parts in the
given congruent triangles.
Given: ∆ 𝐴𝑃𝐸 ≅ ∆𝐴𝐶𝐸
Group 3
List the six pairs of corresponding congruent parts in the
given congruent triangles.
Given: ∆ 𝐶𝑂𝐷 ≅ ∆ 𝑆𝑂𝑌
57
Group 4
List the six pairs of corresponding congruent parts in the
given congruent triangles.
Given: ∆ 𝑃𝑁𝑅 ≅ ∆ 𝑃𝐶𝑅
3. Processing the
answer



4. Reinforcing of the
skills
5. Summarizing the
Lesson
C. Assessment:
D. Agreement/
Assignment:
What are the corresponding congruent parts?
How many pairs of corresponding parts are
congruent if two triangles are congruent?
What are congruent triangles?
Which triangles are congruent if ̅̅̅̅̅
𝑀𝐴 ≅ ̅̅̅̅
𝐾𝐹 , ̅̅̅̅
𝐴𝑋 ≅ ̅̅̅̅
𝐹𝐶, ̅̅̅̅̅
𝑀𝑋 ≅
̅̅̅̅
𝐾𝐶 ; ∠𝑀 ≅ ∠𝐾, ∠𝐴 ≅ ∠𝐹, ∠𝑋 ≅ ∠𝐶. Draw the triangles.


When do we have congruent triangles?
How many pairs of corresponding parts are
congruent if two triangles are congruent?
Given ∆𝑄𝑍𝑃 ≅ ∆𝐾𝑅𝐴. List six pairs of corresponding
congruent parts.
In quadrilateral ABCD, AC bisects ∠𝐷𝐴𝐵 and ∠𝐷𝐶𝐵. Why
are angles B and D congruent?
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
58
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
59
(M8GE-IIId-1) - Illustrates Triangle Congruence
School
Teacher
Grade Level
Learning
Area
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
LC Code
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s
Materials pages
3. Textbook pages
4. Additional
Materials
from
Learning
Resource (LR)
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary
Activities/
Motivation
B. Presentation of the
Lesson
1. Opener
2. Group Activity
Quarter
8
MATHEMATICS
THIRD
Week 4, Day 4
The learner demonstrates understanding of key
concepts of triangle congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner illustrates triangle congruence.
M8GE-IIId-1
Illustrating triangle congruence
383-384
351-353
Mathematics for the 21st Century page 281
Advance Learners
Average Learners
Review:
When do you say that two triangles are congruent?
Cite examples of things, landmarks or structures where
you can see the use of congruent triangles.
List the corresponding congruent parts of the given
congruent triangles and complete the congruence
markings. (By Group)
Group 1: ∆LOT ≅ ∆MES
60
Group 2: ∆BEC ≅ ∆BAC
Group 3: ∆MAP ≅ ∆SPA
Group 4: ∆BOY ≅ ∆NOR
Group 5: ∆GIV ≅ ∆SAV
3. Processing the
Answer





4. Reinforcing the
Skills
What are the two congruent triangles?
What are the corresponding vertices of the two
congruent triangles?
What are the corresponding congruent parts of the
congruent triangles?
How do you identify the corresponding congruent
parts of the congruent triangles?
What do the markings indicate?
Complete the congruence statement.
61
1. ∠G ≅ ____
2. ∠GRN ≅ ______
3. ________ ≅ ∠ENR
4. ____ ≅ ̅̅̅̅
RE
̅̅̅̅
5. ____ = EN
6. ̅̅̅̅
RN≅ ______
7. ∆________ ≅ ∆__________
5.Summarizing the
Lesson



C. Assessment
What are congruent triangles?
How many pairs of corresponding parts are
congruent if two triangles are congruent?
How do you identify the corresponding congruent
parts of congruent triangles?
A. Complete each congruence statement.
1. ∆_______ ≅ ∆_________
2. ∠F ≅ ____
3. ∠A ≅ ______
4. ________ ≅ ∠TIH
5. ____ ≅ ̅̅̅̅
TH
̅̅̅
6. ____ = 𝐼𝑇
̅̅̅≅ ______
7. 𝐹𝐼
B. If ∆HPE ≅ ∆PHO,
statement.
1. ________ ≅ ∠O
2. ∠EHP ≅ ______
3. ________ ≅ ∠OHP
̅̅̅̅
4. ____ ≅ OH
̅̅̅̅
5. HE = _____
6. ̅̅̅̅
HP≅ ______
D. Assignment
V.
REMARKS
VI.
REFLECTION
complete
each
congruence
Given that ∆LKE ≅ ∆USE, identify the congruent
corresponding parts of the two congruent triangles.
62
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which
of
my
teaching strategies
worked well?
F. What difficulties did I
encounter which my
principal
or
supervisor can help
me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
63
(M8GE-IIId-e-1) - Illustrates the SAS Congruence Postulates
School
Grade Level
Learning
Area
Teacher
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Quarter
CONTENT
III.
LEARNING
RESOURCES
References
Teacher’s Guide
pages
Learner’s
pp. 349-372
Materials pages
Textbook pages
2.
3.
MATHEMATICS
THIRD
Week 5, Day 1
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner illustrates the SAS, ASA and SSS
Congruence Postulates
(M8GE – IIId-e-1)
II.
A.
1.
8
Illustrating the SAS Congruence Postulate
4.
Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary
What is an included angle?
(Recall)
Using the figure, name the angles included between the
pairs of sides:
X
1. ̅̅̅̅̅
𝑋𝑀 and ̅̅̅̅̅
𝑁𝑀
L
1
3
̅̅̅̅̅ and 𝑌𝐿
̅̅̅̅
2. 𝑌𝑀
M
̅̅̅̅̅ and 𝑌𝑀
̅̅̅̅̅
3. 𝑋𝑀
Z
̅̅
4. ̅̅̅̅
𝑋𝑍 and ̅̅
𝐿𝑍
2 4
N
5. ̅̅̅̅
𝑌𝑍 and ̅̅̅̅
𝑁𝑍
Y
64
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
Activity
Given equilateral ∆𝑀𝑂𝑁 with perpendicular bisector 𝑂𝑋.
(Note: The teacher must review the concepts of equilateral
triangles and perpendicular bisector.)
O
M
2. Processing the
Answer
N
X

Which corresponding parts of the two triangles are
congruent?
(Note: Prepare a cut-out of an equilateral triangle and
support the student’s answer by folding this triangle
along the perpendicular bisector.
 Are the two triangles congruent? Why?
 How many parts of these two triangles where
identified as congruent?
 How do you describe the angles identified as
congruent?
 State the SAS Congruence Postulate.
(The teacher will explain the SAS Congruence Postulate
“If two sides and the included angle of one triangle are
congruent respectively to the two sides and the included
angles of another triangle, then the two triangles are
congruent.”)
D

Given the figure:
C
̅̅̅̅
𝐷𝐶 ≅ ̅̅̅̅
𝐵𝐶
∠1 ≅ ∠2
1
2
A
B
3. Reinforcing the skills
Find the other pairs of congruent sides.
Supply the missing congruent parts of the following pairs
of triangles to make it congruent
T by SAS.
R
N
1. ∆𝑀𝑂𝑁 ≅ ∆𝑅𝑆𝑇
̅̅̅̅̅
𝑀𝑁 ≅ ̅̅̅̅
𝑅𝑆
̅̅̅̅̅ ≅ _____
𝑀𝑂
S
M
O
∠𝑀 ≅ ______
D
C
2. ∆𝑀𝑂𝑁 ≅ ∆𝑅𝑆𝑇
̅̅̅̅
𝐴𝐵 ≅ ______
∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵
̅̅̅̅
𝐵𝐷 ≅ ______
65
A
B
4. Summarizing the
Lesson
C. Assessment
How do you illustrate SAS congruence postulate?
The pairs of triangles were similarly marked. Write the
three pairs of corresponding congruent parts to show SAS
congruence postulate.
1. C
2. I
J
B
A
O
D
H
K
D. Assignment
1. What is an included side?
2. How do you determine the included side of two
given pairs of angles?
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
66
(M8GE-IIId-e-1) - Illustrates the ASA Congruence Postulates
School
Grade Level
Learning
Area
Teacher
Time &
Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner illustrates the SAS, ASA and SSS
Congruence Postulates
(M8GE – IIId-e-1)
CONTENT
III.
LEARNING
RESOURCES
References
Teacher’s Guide
pages
Learner’s
pp. 349-372
Materials pages
Textbook pages
2.
3.
MATHEMATICS
THIRD
Week 5, Day 2
Quarter
II.
A.
1.
8
Illustrating the ASA Congruence Postulate
4.
Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary
What is an included side?
(Recall)
How do you determine the included side between the given
pairs of angles?
In the figure, name the sides included between the given
pairs of angles:
X
L
1. ∠𝑋 and ∠1
1 3
M
2. ∠𝑦 and ∠2
3. ∠𝑋 and ∠𝑀
2 4
N
4. ∠𝑌 and ∠𝑀
5. ∠𝑋 and ∠4
67
Y
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
Activity
Given equilateral ∆𝑀𝑂𝑁, draw angle bisector 𝑂𝑋.
O
N
X
 Which is the included side of ∠𝑀 and ∠𝑀𝑂𝑋? of ∠𝑁
and ∠𝑁𝑂𝑋?
 What can you say about the measures of ∠𝑀? ∠𝑁?
Why?
 How about the measures of ∠𝑀𝑂𝑋 and ∠𝑁𝑂𝑋?
Why?
̅̅̅̅̅ and
 What can you say about the lengths of 𝑀𝑂
̅̅̅̅
𝑁𝑂 ? Why?
 Do you think ∆𝑀𝑂𝑋 ≅ ∆𝑁𝑂𝑋? How can you prove
that?
 What are the congruent parts of the two triangles?
 State the ASA Congruence Postulate.
(The teacher will explain the ASA Congruence Postulate
“If two angles and the included side of one triangle are
congruent respectively to the two angles and the included
side of another triangle, then the two triangles are
A
congruent.”)
 Given the figure:
1
3
F
Y
2
4
∠1 ≅ ∠2
∠3 ≅ ∠4
E
Find the other congruent parts. What are the congruent
triangles?
Supply the missing congruent parts of the following pairs
of triangles in order that it will be congruent by ASA.
M
2. Processing the
Answer
3. Reinforcing the skills
1. ∆𝑃𝑂𝑌 ≅ ∆𝑀𝐼𝐾
∠𝑂 ≅ ∠𝐼
∠𝑌 ≅ _____
̅̅̅̅
𝑂𝑌 ≅ ______
P
K
I
2. ∆𝐾𝐴𝑇 ≅ ∆𝑇𝐸𝐾
∠𝐴𝐾𝑇 ≅ ∠𝐸𝑇𝐾
∠𝐴𝑇𝐾 ≅ _______
E
68
O
K
M
̅̅̅̅
𝐾𝑇 ≅ ______
T
Y
A
4. Summarizing the
Lesson
C. Assessment
How do you illustrate ASA congruence postulate?
Illustrate ASA Congruence Postulate
The pairs of triangles were similarly marked. Which 3 pairs
of corresponding parts must be congruent so that the
triangles are congruent by ASA congruence postulate?
A
L
1.
2.
12
O
E
Y
D
1 4
E
3 4
E
H
3.
G
D. Assignment
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
3 2
F
Study the SSS congruence postulate.
69
T
(M8GE-IIId-e-1) - Illustrates the SSS Congruence Postulates
School
Grade Level
Learning
Area
Teacher
Time &
Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
Quarter
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner illustrates the SAS, ASA and SSS
Congruence Postulates
(M8GE – IIId-e-1)
II.
CONTENT
III.
LEARNING
RESOURCES
References
Teacher’s Guide
pages
Learner’s
pp. 349-372
Materials pages
Textbook pages
A.
1.
2.
3.
Additional
Materials
from
Learning
Resource
(LR)
portal
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary
(Recall)
8
MATHEMATIC
S
THIRD
Week 5, Day 3
Illustrating the SSS Congruence Postulate
4.
The teacher will bring the following materials on the class:
a. 5 pcs. of scissors (scissors for elem.)
b. 10 used folders
c. 5 pcs. of rulers
d. 5 pcs. of pentel pen
Direction: Group the students into 5 groups. Then the
students will perform the following:
1.
Ask each group to cut out two triangles whose
sides measure: 6in, 8in, 10in and 5in, 12in, 13in.
2.
Indicate the measures along the sides of the
triangles.
70
B. Presentation of the
Lesson
1. Problem Opener/
Group Activity
Activity:
- Paste the triangles with the same measures of the
sides on the same side.
- Are the group of triangles congruent? How can you
show it?
- Assign letters to corresponding vertices to name
each pair of triangles.
- Identify each pair of corresponding sides of each
pair of triangles.
 Which part of these congruent triangles were
measured?
 Are these measured parts coincide with each other
when two of these triangles made to coincide?
 If letters were assigned to each vertex, write the
corresponding congruent parts of the two triangles.
 State your findings in a sentence. What do you call
this statement?
Put similar marks on the corresponding congruent sides
to make the pairs of triangles congruent by SSS
Congruence Postulate.
a.
C
b.
F
G
2. Processing the
Answer
3. Reinforcing the skills
B
c.
G
P
4. Summarizing the
Lesson
C. Assessment
I
E
D
H
T
R
U
S
How do you illustrate SSS congruence postulate?
Illustrate SSS Congruence.
1.
Put
similar 2. Make a list of the 3. Explain why the
markings on the corresponding
two triangles are
two triangles to congruent sides of congruent.
show that the two the two triangles.
triangles
are
congruent by SSS.
A
M
T
L
B F
E
X
C
G
A E
71
O
V
H
D. Assignment
PQRS is a rhombus. If 𝑄𝑅 = 12𝑐𝑚,
1. how longis PS?
2. which side corresponds to RS?
3. Prove: ∆𝑃𝑄𝑆 ≅ ∆𝑅𝑆𝑄.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to share
with other teachers?
72
Q
P
R
S
(M8GE-IIIe-1) - Illustrates the SAS, ASA and SSS Congruence
Postulates
School:
Teacher:
Grade Level:
Learning
Area:
Quarter:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary Activity:
8
MATHEMATICS
THIRD
Week 5, Day 4
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations
The learner illustrates the SAS, ASA and SSS Congruence
Postulates
(M8GE – IIId-e-1)
Illustrating the SAS, ASA, and SSS Congruence Postulate
380-392
292-294
Moving Ahead with Mathematics II, 1999 (pages 114-115)
Advanced Learners
Average Learners
By what triangle congruence postulate are the following
triangles said to be congruent?
1.
73
B. Presentation of the
Lesson
1. Problem Opener:
2. Group Activity
What corresponding parts should be congruent so that the
two triangles are congruent?
The class will be divided into 3 groups to perform an
activity
Activity:
Corresponding congruent parts are marked. Indicate
the additional corresponding parts needed to make the
triangles congruent by using the specified congruence
postulates. Discuss the answer to the class.
Group 1:
Group 2:
Group 3:
3. Processing the
answer
4. Reinforcement of the
skill
5. Summarizing the
Lesson
 What are the additional parts of the triangles should
be congruent?
 At least there will be how many corresponding part
should be congruent so that the two triangles are
congruent?
∆ 𝐴𝐵𝐶 ≅ ∆𝐷𝑂𝑇 by SSS Congruence. If m AB = 2cm, m
OT = 8 cm, and m AC = 7 cm., what is the measure of side
BC and DT?

At least there how many corresponding part should
be congruent so that the two triangles are
congruent?
74

C. Assessment:
D. Agreement/
Assignment:
Illustrate triangle congruence.
What additional information is needed to have the SAS
congruence postulate so that ∆ 𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 if 𝐴𝐵 ≅ DF
and ∠𝐵 ≅ ∠𝐷? Draw the triangles.
ΔABC ≅ ΔDEF, which segment is congruent to AB:
a.
b.
c.
d.
BC
AC
DE
EB
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
75
(M8GE-IIIf-1) - Solves Corresponding Parts of
Congruent Triangles
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
8
MATHEMATICS
THIRD
Week 6, Day 1
The learner demonstrates understanding of key concepts
of triangle congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner solves corresponding parts of congruent
triangles
M8GE-IIIf-1
Solving corresponding parts of congruent triangles
C. Learning
Competencies/
Objectives
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide 383-384
pages
2. Learner’s Materials 351-353
pages
3. Textbook pages
Geometry III.2009. pp. 89-91
Mathematics for the 21st Century page 278-282
4. Additional
Materials
from
Learning Resource
(LR)
B. Other
Learning National Training of Trainers on Critical Content in
Resources
Mathematics 8 Material
IV.
PROCEDURE
Advance Learners
Average Learners
A. Preliminary Activities/
Review:
Motivation
When do you say that two triangles are congruent?
B. Presentation of the
Lesson
1. Opener
Name the two congruent triangles.
List the 6 corresponding congruent parts.
76
2. Group Activity/
Presentation
If ∆BIN ≅ ∆CKS , find the indicated measures by group.
C
I
11
9
52°
12
K
83°
N
B
S
Group 1:
BN = ________
Group 2:
KS = ________
Group 3 : m∠B = ________
Group 4:
m∠I = __________
Group 5: Perimeter of ∆BIN = ________
3. Processing the
Answer









4. Reinforcing the Skills
What are the two congruent triangles?
What are the corresponding vertices of the two
congruent triangles?
̅̅̅ is congruent to what side of ∆CKS? Then, what is
𝐵𝐼
BI?
̅̅
̅̅ is congruent to what side of ∆CKS? Then, what is
KS
KS?
∠B is congruent to what angle of ∆CKS? Then, what
is m∠B?
∠I is congruent to what angle of ∆CKS? Then, what
is m∠I?
What are the lengths of the sides of ∆BIN?
What is the perimeter of ∆BIN? ∆CKS?
How do you find the unknown length of sides or
measure of angles of two congruent triangles?
If ∆𝐴𝐶𝑇 ≅ ∆𝐿𝐸𝑆, complete the congruence statement or find
the indicated measure.
S
L
1. ∠C ≅ ____
40°
2. m∠C = ____
3. ____ ≅ CT
4. ____ = 9 units
5
5. ∠L ≅ ____
A
6. m∠L = ______
7. AT ≅ _____
8. ____= 5 units
E
U
11
93°
9
T
77
C
5.Summarizing the
Lesson
C. Assessment


What are congruent triangles?
How many pairs of corresponding parts are
congruent if two triangles are congruent?
If ∆PIE ≅ ∆TRY, complete each congruency statement
below.
I
T
1. ∠E ≅ ____ P
2. ∠I ≅ ____
Y
3. ̅̅̅̅
PE ≅ ____
4. ̅̅̅̅
RY ≅ ____
5. ̅̅̅̅
TR ≅ ____
E
6. ∠𝑃 ≅ ____
7. If 𝑅𝑌 = 10, then EI = _____
R
8. If 𝑚∠𝑌 = 76, then 𝑚∠𝐸 = _____
9. If 𝑚∠𝑇 = 60 and 𝑚∠𝐼 = 65, then ∠𝑌 = ____
10. If IP = 17, PE = 16 and EI= 15, what is the perimeter
of ∆𝑇𝑅𝑌?
Note: The triangles are not drawn to scale.
D. Assignment
V.
Name the congruent triangles and the corresponding
congruent parts.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
78
E. Which of my teaching
strategies
worked
well?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized
material/s
did I use/discover
which I wish to share
with other teachers?
79
(M8GE-IIIf-1) - Solves Corresponding Parts of Congruent
Triangles
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
B. Motivation/
Preliminary Activity:
B. Presentation of the
Lesson
1. Problem Opener:
2. Group Activity
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 6, Day 2
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations
The learner solves corresponding parts of congruent
triangles.
(M8GE-IIIf-1 )
Solving corresponding parts of congruent triangles
380-392
292-294
Moving Ahead with Mathematics II, 1999 (pages 114-115)
Advanced Learners
Average Learners
What are the different triangle congruence postulates being
used to prove that the two triangles are congruent?
Can we find the measure of the corresponding parts of
congruent triangles by using the different triangle
congruence postulates?
The class will be divided into 4 groups to perform an
activity.
Activity:
Find the value of x in the two congruent triangles.
Write your solutions and explain your answer to the class.
80
Learning Group 1:
Learning Group 2:
Learning Group 3:
Find the value of x and y.
Learning Group 4:
3. Processing the
answer
 What are the congruent triangles?
 By what triangle congruence postulates is used to
prove that the two triangles are congruent?
 What are the corresponding congruent parts of
congruent triangles?
81
 How do we solve for the unknown parts of congruent
triangles?
4. Reinforcement of the
skill
Find the value of x and the unknown measure of angles.
800
5. Summarizing the
Lesson

How did you find the measure of the corresponding
parts of congruent triangles?
Find the value of x in the two congruent triangles.
C. Assessment:
D. Agreement/
Assignment:
What are the triangle congruence theorems in a right
triangle?
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
82
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
83
(M8GE-IIIf-1) - Solves Corresponding Parts of Congruent
Triangles
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional
Materials
from
Learning Resource
(LR)
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary Activities/
Motivation
B. Presentation of the
Lesson
1. Opener
2. Group Activity/
Presentation
GRADE 8
MATHEMATICS
THIRD
Week 6, Day 3
The learner demonstrates understanding of key
concepts of triangle congruence.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations.
The learner solves corresponding parts of congruent
triangles.
M8GE-IIIf-1
Solving corresponding parts of congruent triangles
383-384
351-353
Mathematics for the 21st Century page 280-282
National Training of Trainers on Critical Content in
Mathematics 8 Material
Advanced Learners
Average Learners
Review:
Recite and explain the acronym CPCTC.
Given ∆MNP ≅ ∆QRS, complete the congruence
statement in each item.
1. ∠Q ≅ _____
2. ∠P ≅ _____
3. ∠R ≅ _____
̅̅ ≅ _____
4. ̅̅
SR
̅̅̅̅ ≅ _____
5. MP
6. ̅̅̅̅̅
MN ≅ _____
Given 𝐺𝐸𝑂  𝑇𝑅𝐼 and 𝑚𝑂 = 30, 𝑚𝑅 = 100, 𝑇𝐼 =
11,
RI =7 and GE= 5. Find the unknown by group.
84
3. Processing the
Answer
4. Reinforcing the Skills
Group 1: EO and GO
Group 2: m∠E and m∠I
Group 3: Perimeter of GEO
Group 4: Perimeter of TRI
 What are the two congruent triangles?
 What are the corresponding vertices of the two
congruent triangles?
 ̅̅̅̅
EO is congruent to what side of ∆TRI? Then, what
is EO?
̅̅̅̅ is congruent to what side of ∆TRI? Then, what
 GO
is GO?
 ∠E is congruent to what angle of ∆TRI? Then,
what is m∠E?
 ∠I is congruent to what angle of ∆GEO? Then,
what is m∠I?
 ∠G is congruent to what angle of ∆TRI? Then,
what is m∠G?
 How do you solve for the m∠G?
 What are the lengths of the sides of ∆GEO? ∆TRI?
 What is the perimeter of ∆GEO? ∆TRI?
 How will you compare the perimeters of the two
triangles?
 How do you find the unknown length of sides or
measure of angles of two congruent triangles?
1. ΔQED ≅ ΔCAT, QE = 9x, ED = 4x+3, DQ = 5x+2, and
AT = x+9. Find AC and CT.
2. Triangles ABC and DEF are congruent. If AB = DE,
BC = EF, ABC  37 and EDF  39 , what is the
measure of EFD ?
5.Summarizing the
Lesson
C. Assessment


What are congruent triangles?
How do you find the measure of unknown parts
of 2 congruent triangles?
TOM  VER and mT = 83, mR=32, TM = 10,
RE =14.Complete the congruence statements and find
the indicated measures
1.
2.
3.
4.
5.
6.
7.
8.
TO  _____
O  _____
mM= _____
OM = _____
VE  _____
mE = _____
RV = _____
T _____
85
9-10. If ∆ABC ≅ ∆PQR , AB = 8x, BC = 5x - 1, PR = 2x+3,
and PQ = 4x + 4. Find AB and PR.
D. Assignment
V.
Triangles ABC and DEF are congruent. If AB = DE, BC
= EF, ABC  37 and EDF  39 , what is the
measure of EFD ?
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities for remediation.
C. Did the remedial lessons
work? No. of learners
who have caught up with
the lesson.
D. No. of learners who
continue
to
require
remediation.
E. Which of my teaching
strategies worked well?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What
innovation
or
localized material/s did I
use/discover which I
wish to share with other
teachers?
86
(M8GE-IIIf-1) - Solves Corresponding Parts of Congruent
Triangles (Performance Task)
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
Presentation of the
Lesson
Grade Level:
Learning Area:
8
MATHEMATICS
Quarter:
THIRD, Week 6 Day 4
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations
The learner solves corresponding parts of congruent
triangles. (Performance Task)
(M8GE-IIIf-1 )
Solving corresponding parts of congruent triangles
380-392
292-294
Moving Ahead with Mathematics II, 1999 (pages 114-115)
Advanced Learners
Average Learners

Reviewing previous lessons
As Grade 8 students they need to apply the
learning to real life situations.
 Establishing the purpose for the lesson.
The grade 8 students will be given a practical task
which will demonstrate their understanding in
Triangle Congruence
 Evaluating learning
The performance task for 50 points will follow
the GRASPS model.
GOAL
To make a design or blueprint of a bridge suspension.
87
ROLE
AUDIENCE
As one of the engineers of the DPWH who is
commissioned by the Special Project Committee.


SITUATION
City Council together with the City Engineers.
(Classmates/ group mates)
Teacher
One of the moves of the City Council for economic
development is to connect a nearby island to the mainland
with a suspension bridge for easy accessibility of the people.
Those from the island can deliver their produce and those
from the mainland can enjoy the beautiful scenery and
beaches of the island.
Suppose you are one of the Engineers of the DPWH
who is commissioned by the Special Project Committee to
present design/blueprint of a suspension bridge to the City
Council. How would you design/blueprint look like? How
would you convince the City Council that the design is stable
and strong.
PERFORMANCE/
PRODUCT
STANDARD FOR
GRADING/ CRITERIA
a design/blueprint of a suspension bridge
See the attached rubric for scoring.
(see also Mathematics 8 LM, page 370)
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
88
RUBRIC FOR SUSPENSION BRIDGE DESIGN/BLUEPRINT
89
(M8GE-IIIg-1) - Proves Two Triangles are Congruent
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content
Standards:
B. Performance
Standards:
C. Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Guide Pages
3. Textbook
Pages
4. Additional
Material from
Learning
Resource
Material
B. Other
Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary
Activity
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 7, Day 1
The learner demonstrates understanding of key concepts of
axiomatic structure of geometry and triangle congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving congruent
triangles using appropriate and accurate representations.
The learner proves that two triangles are congruent by SSS
postulate.
(M8GE-IIIg-1)
Proving Congruence of Triangles (SSS Postulate)
389-385
359-361
Grade 8 Mathematics Patterns and Practicalities by Gladys C.
Nivera, Ph.D. pp.368-369
Advanced Learners
Average Learners
Recall and observe that the Properties of congruence for
triangles are similar to the Properties of Equality for real
numbers.
 Reflexive Property: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐵 ; ∠𝐶 ≅ ∠𝐶
 Symmetric Property: If ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐸 , then ̅̅̅̅
𝐷𝐸 ≅ ̅̅̅̅
𝐴𝐵 .
If ∠𝐴 ≅ ∠𝐵, then ∠𝐵 ≅ ∠𝐴.
 Transitive Property: If ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐷𝐸 and ̅̅̅̅
𝐷𝐸 ≅ ̅̅̅̅
𝐺𝐻 , then
̅̅̅̅
̅̅̅̅
𝐴𝐵 ≅ 𝐺𝐻.
If ∠𝐴 ≅ ∠𝐵, and ∠𝐴 ≅ ∠𝐵 ≅ ∠𝐶,
then ∠𝐴 ≅ ∠𝐶.
90
B. Presentation
of the
Lesson

Get three sticks of unequal lengths label it p, q, r.
p
q
r

How many different triangles can you form using the three
sticks?
1. Problem
Opener:
Consider the triangles below,



2.Group Activity
Name the corresponding parts of the two triangles
shown.
Are the corresponding parts of the two triangles
congruent?
When the corresponding sides of two triangles are
congruent, does it necessarily follow that the two
triangles
are congruent?
Why or why not?
Given: Square LOVE and its diagonal ̅̅̅̅
𝐿𝑉
91
3.Processing the
answer



What are the given congruent parts of the two triangles?
How will you show your proof using the two – column form?
What do you mean by SSS Postulate?
Note: Side-Side-Side (SSS) Postulate
If three sides of one triangle are congruent to three
sides of another triangle, then the two triangles are congruent.
4.Reinforcement of
the skill
5.Summarizing the
Lesson
C. Assessment:
Given:
1. What is the SSS Congruence Postulate?
2.How do you prove that two triangles are congruent by SSS
Postulate?
Given: Rhombus PRAY and its diagonal ̅̅̅̅
𝑅𝑌
Prove:
92
D. Agreement/
Assignment:
Draw a figure and mark the given information. Then state what
is given and what is to be proved.
A diagonal ̅̅̅̅
𝐵𝐴 is drawn on rectangle BOAT.
Prove that the two triangles formed by a diagonal are
congruent.
V. REMARKS:
VI. REFLECTION:
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional
activities for
remediation who
scored below
80%
C. Did the remedial
lesson work? No.
of learners who
caught up with
the lesson
D. No. of learners
who continue to
require
remediation.
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/discover
which I wish to
93
(M8GE-IIIg-1) - Proves Two Triangles are Congruent
School:
Teacher:
Time and
Date:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 7, Day 2
A. Content
Standards:
B. Performance
Standards:
The learner demonstrates understanding of key concepts of
triangle congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving congruent
triangles using appropriate and accurate representations.
C. Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Guide Pages
3. Textbook
Pages
4. Additional
Material from
Learning
Resource
Material
B. Other
Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary
Activity
The learner proves that two triangles are congruent by SAS
postulate.
(M8GE-IIIg-1)
Proving Congruence of Triangles (SAS Postulate)
B. Presentation of
the Lesson
389-395
358-364
Moving ahead with Mathematics II, 1999, pp. 121-123
Geometry III, pp. 98-100
Advanced Learners
Average Learners
 What are congruent triangles?
 Can you give the different triangle congruence
postulates?
 Can you recall the meaning of the following terms?
a. Midpoint of a line segment
b. Vertical angles
c. Perpendicular Lines
 With your knowledge of the definition of congruent
triangles and the different triangle congruence postulates
are you ready to prove the congruence of two triangles?
94
1. Problem
Opener/
Activity
LET’S DO IT
Directions: Find out how you can apply the Congruence
Postulates to prove that two triangles are congruent.
̅̅̅̅ ≅ 𝐷𝐸
̅̅̅̅
Given: 𝐴𝐵
∠𝐵 ≅ ∠𝐸
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐸𝐹
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
2. Processing the
answer
 What are the given congruent parts of the two triangles?
 Can you show the proof using the two-column form?
 What do you mean by SAS postulate?
Note:
The Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of another
triangle, then the two triangles are congruent.
 What is an included angle of a triangle?
An included angle of a triangle is the angle formed by the
common vertex of the two sides of the triangle.
3. Reinforcement of
the skill
4. Summarizing the
Lesson
1.
When can you say that two triangles are congruent by
SAS?
2. How do you prove that two triangles are congruent using
SAS Postulate?
95
C. Assessment:
̅̅̅̅ ⊥ 𝐶𝐷
̅̅̅̅
In the figure, 𝐵𝐴
A is the midpoint of ̅̅̅̅
𝐶𝐷
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐵𝐷
C
D. Agreement/
Assignment:
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
96
B
A
D
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
97
(M8GE-IIIg-1) Proves Two Triangles are Congruent
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content
Standards:
B. Performance
Standards:
C. Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Guide Pages
3. Textbook
Pages
4. Additional
Material from
Learning
Resource
Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary
Activity
B. Presentation of
the Lesson
1. Problem
Opener:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 7, DAY 3
The learner demonstrates understanding of key concepts of
triangle congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving congruent
triangles using appropriate and accurate representations.
The learner proves that two triangles are congruent by ASA
postulate.
(M8GE-IIIg-1)
Proving two triangles are congruent (SSS Postulate)
389-385
360-361
Grade 8 Mathematics Patterns and Practicalities by Gladys C.
Nivera, Ph.D. pp.368-371
Advanced Learners
Average Learners
 Draw a triangle with these measures; two angles
measure 500 and 600 and the length of the included
side is 5 cm.
 How many different triangles having these measures
can you draw?

What conjecture about triangle congruence can you
make based on this activity?
98
2.Group Activity
3. Processing the
answer
 What are the given congruent parts of the two triangles?
 How will you show your proof using the two – column form?
 What do you mean by ASA Postulate?
Note: Angle-Side-Angle (ASA) Postulate
If two angles and the included side of one triangle are
congruent to two angles and included side of another triangle,
then the two triangles are congruent.
4. Reinforcement of
the skill
5. Summarizing the
Lesson
C. Assessment
1.
How do you prove that two triangles are congruent using
ASA Congruence Postulate?
2.
When can you say that two triangles are congruent by ASA
Postulate?
Direction: Fill in the missing reasons of the two-column proof to
prove that the two triangles are congruent.
̅̅̅̅ ;
Given: 𝐶 is the midpoint of 𝐴𝐸
∠𝐴 ≅ ∠𝐸
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐷𝐶
99
D. Agreement/
Assignment:
Given: ∆𝐻𝑂𝑇 and ∆𝑀𝐸𝑇;
T is a midpoint of ̅̅̅̅
𝑂𝐸 ;
∠𝑂 and ∠𝐸 are right angles
Prove: ∆𝐻𝑂𝑇 and
H
O
∆𝑀𝐸𝑇
T
E
M
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for
remediation who
scored below 80%
C. Did the remedial
lesson work? No. of
learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked
well? Why did these
work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did
I use/discover which I
wish to share with
other teachers?
100
(M8GE-IIIg-1) - Proves Two Triangles are Congruent
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content
Standards:
B. Performance
Standards:
C. Learning
Competency:
LC Code:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s
Guide Pages
2. Learner’s
Guide Pages
3. Textbook
Pages
4. Additional
Material from
Learning
Resource
Material
B. Other
Learning
Resources
IV. PROCEDURES:
A. Motivation/
Preliminary
Activity
B. Presentation of
the Lesson
1. Problem
Opener:
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 7, Day 4
The learner demonstrates understanding of key concepts of
triangle congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving congruent
triangles using appropriate and accurate representations
The learner proves that two triangles are congruent by SAA/
AAS postulate.
(M8GE-IIIg-1 )
Proving that two triangles are congruent
389-385
360-361
Grade 8 Mathematics Patterns and Practicalities by Gladys C.
Nivera, Ph.D. pp.368-371
Advance Learners
Average Learners
 Draw a triangle with these measures ;
two angles
measure 400 and 600 and the length of the side opposite
one of the angles is 4 cm.
 How many different triangles having these measures can
you draw?

 If two angles and a side opposite one of them in one
triangle are congruent to the corresponding angles and
side in another triangle, does it necessarily follow that
the two triangles are congruent?
Why or why not?
101
2.Group Activity
Given: ∆𝐵𝐿𝑈 is an isosceles triangle.
̅̅̅̅
𝐿𝐸 is perpendicular to ̅̅̅̅
𝐵𝑈
Prove:
L
∆𝐵𝐿𝐸 ≅ ∆𝑈𝐿𝐸
B
3. Processing the
answer

U
E
What are the given congruent parts of the two triangles?

How will you show your proof using the two – column form?
Statements
Reasons
1. ∆𝐵𝐿𝑈 is an isosceles
triangle
2. 𝐵𝐿 ≅ 𝐿𝑈
1.Given
∠𝐵 ≅ ∠𝑈
3.
4. ̅̅̅̅
𝐿𝐸 is perpendicular to ̅̅̅̅
𝐵𝑈
5. ∠𝐵𝐸𝐿 and ∠𝑈𝐸𝐿 are
right angles
6. ∠𝐵𝐸𝐿 ≅ ∠𝑈𝐸𝐿
∆𝐵𝐿𝐸 ≅ ∆𝑈𝐿𝐸
7.
2. Definition of Isosceles
Triangle
3. Base angles of an
isosceles triangle are
congruent.
4. Given
5.Definition of
perpendicularity
6. Right angles are congruent
7. SAA Postulate
 What do you mean by SAA Postulate?
Note:
Side-Angle-Angle(SAA) Postulate / Angle-Angle Side
(AAS) Postulate
If two angles of a triangle and a side opposite one of its
angles are congruent to two angles and a side opposite one of
the angles of another triangle, then the two triangles are
congruent.
4. Reinforcement of
the skill
Given the figure below, identify the congruent parts to prove
that the two triangles are congruent by SAA / AAS postulate.
X
A
C
________________
________________
________________
∆𝐵𝑂𝑋 ≅ ______
B
O
102
R
5.Summarizing the
Lesson
C. Assessment:
1.
How do you prove that two triangles are congruent by
SAA Congruence Postulates?
For each figure below, give the congruent parts to prove that the
two triangles are congruent by SAA / AAS postulate.
S
L
Fig.1
G
A O
I
1. ______________
2. ______________
3. ______________
4.
Fig. 2
GAS ≅ ________
F
E
A
D
5. _______________
6. _______________
7. _______________
8. ________ ≅
D. Agreement/
Assignment
DAF
̅̅̅ is the perpendicular
In an isosceles triangle FIX, 𝐼𝑇
̅̅̅̅
bisector of 𝐹𝑋.
Prove that the two triangles formed by the perpendicular
bisector are congruent.
V. REMARKS:
VI. REFLECTION:
A. No. of learners
who earned 80%
in the evaluation
B. No. of learners
who require
additional
activities for
remediation who
scored below
80%
103
C. Did the remedial
lesson work? No.
of learners who
caught up with
the lesson
D. No. of learners
who continue to
require
remediation.
E. Which of my
teaching
strategies worked
well? Why did
these work?
F. What difficulties
did I encounter
which my
principal or
supervisor can
help me solve?
G. What innovation
or localized
materials did I
use/discover
which I wish to
share with other
teachers?
104
(M8GE-IIIh-1) - Proves Statements on Triangle Congruence
School:
Teacher:
Grade Level:
Learning
Area:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance Standards:
C. Learning Competencies:
II. CONTENT:
Quarter:
8
MATHEMATICS
THIRD
Week 8, Day 1
The learner demonstrates understanding of key
concepts of triangle congruence.
The learners shall be able to communicate
mathematical thinking with coherence and clarity in
formulating, investigating, analyzing, and solving
real-life problems involving congruent triangles using
appropriate and accurate representations.
The learner proves statements on triangle
congruence using HA Congruence Theorem.
(M8GE-IIIh-1 )
Proving Right Triangle Congruence by HA
(Hypotenuse- Acute angle Congruence Theorem)
III. LEARNING RESOURCES:
A. References
1. Teacher’s Guide Pages
pp. 392-393
2. Learner’s Guide Pages
3. Textbook Pages
4. Additional Material from
Learning Resource
Material
B. Other Learning
Resources
pp. 361-362
Patterns and Practicalities by Gladys C. Nivera pp.
387-388
Advanced Learners
IV. PROCEDURE
A. Preliminary Activity/
Motivation
Average Learners
Theorem Relay!
Instruction:
The class will be divided into 5 groups. They will
form a line (can be done inside /outside the
classroom). The leader will determine which theorem
applies to prove that the two triangles are congruent.
Then the leader will pass the theorem (message) to
the next up to the last member. The last member will
give the answer to the teacher. The fastest to
complete the relay will be the winner.
Use the figures below.
105

B. Presentation of the
Lesson
1. Problem Opener
What are the different triangle congruence
theorems mentioned?
 How do we know that the two triangles can be
proved by SSS, SAS, ASA?
Tabaco City is known for being the City of Love,
thus many visitors come to this City. One of the
most visited places in Tabaco is the City plaza.
Kiosk of Plaza
Let us assume that the side of the kiosk in the plaza
are two right triangles. Let us call them, ∆𝐴𝑇𝐶 and
∆𝑁𝑊𝑂, where angles T and W are right angles. Let
us try to find out if these two triangles are congruent.
106
Given: ∠𝐴 ≅ ∠𝑁
̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝑁𝑂
Prove: ∆𝐴𝑇𝐶 ≅ ∆𝑁𝑊𝑂
Proof:
(Selected students will be given piece of paper
containing statement and reason. On the board they
are going to paste the statement and the reason
given to them.
Statements
Reasons
∆𝐴𝑇𝐶 ≅ ∆𝑁𝑊𝑂
∠𝐴 ≅ ∠𝑁
̅̅̅̅ ≅ ̅̅̅̅
𝐴𝐶
𝑁𝑂
Given
∠𝑇 ≅ ∠𝑊
All right angles are congruent.
Given
* What theorem can we use to prove that the two
triangles are congruent? Is it SSS, SAS, ASA,
SAA?
* If ∆𝐴𝑇𝐶 and ∆𝑁𝑊𝑂 are right triangles, how do you
call sides AC and NO of the said right triangles?
* Comparing angles A and N to angles T and W, if T
and W are right angles, how do you call angles A
and N?
* Therefore, how are ∆𝐴𝑇𝐶 𝑎𝑛𝑑 ∆𝑁𝑊𝑂 congruent?
What particular parts are being considered?
* Then, ∆𝐴𝑇𝐶 ≅ ∆𝑁𝑊𝑂 by the Triangle
Congruence Theorem called, Hypotenuse and
Acute Angle Congruence Theorem (HA)
*Introduce to them that AAS theorem in right
triangle is either HA (Hypotenuse-Acute angle)
Theorem or LA (Leg-Angle) Theorem.
107
2. Group Activity
To understand better the HA Theorem, the class will
be grouped into 4. Groups 1 and 2 will be given
same two triangles to prove as with groups 3 and 4.
Supply the missing statements/reasons to complete
the proof.
For Groups 1 and 2
Given: ∠𝐴 𝑎𝑛𝑑 ∠ 𝐸 are right angles
̅̅̅̅
𝐹𝑇 bisects ∠𝐴𝑇𝐸
Prove:∆𝐹𝐴𝑇 ≅ ∆𝐹𝐸𝑇
Proof:
Statements
∠𝐴 𝑎𝑛𝑑 ∠𝐸 are right angles
Reasons
All right angles are
congruent
̅̅̅̅
𝐹𝑇 bisects ∠𝐴𝑇𝐸
∠𝐴𝑇𝐹 ≅ ∠𝐸𝑇𝐹
Given
Reflexive Property
∆𝐹𝐴𝑇 ≅ ∆𝐹𝐸𝑇
For Groups 3 and 4
Given: 𝑅𝐸𝐴𝐿 is a rectangle
∠𝐿𝑅𝐴 ≅ ∠𝐸𝐴𝑅
Prove: ∆𝑅𝐿𝐴 ≅ ∆𝐴𝐸𝑅
Proof:
Statements
Reasons
∠𝐿𝑅𝐴 ≅ ∠𝐸𝐴𝑅
Reflexive property
𝑅𝐸𝐴𝐿 is a rectangle
∠𝑅𝐿𝐸 𝑎𝑛𝑑
∠𝐴𝐸𝑅 are right
angles
Definition of right
triangles
∆𝑅𝐿𝐴 ≅ ∆𝐴𝐸𝑅
**A representative of each group will present their
work.
108
3. Processing the
Answers
1. From the name HA theorem, what parts of
the triangle must be proved congruent first
before saying that the two triangles are
congruent by HA theorem?
2. What kind of triangles are we proving using
the HA congruence triangle theorem?
3. What is the difference of Hyl theorem to HA
theorem?
What about their similarities?
4. Do you think, acute and obtuse triangles can
also be proved by HA triangle congruence
theorem?
4. Reinforcing the Skills
Given: ∆𝐸𝑃𝑈 is an isosceles triangle
̅̅̅̅
𝑃𝑅 bisects ∠𝐸𝑃𝑈
∠𝑃𝑅𝑈 and ∠𝑃𝑅𝐸 are right angles
Prove: ∆𝑃𝑅𝐸 ≅ ∆𝑃𝑅𝑈
Proof:
Statements
Reasons
∆𝑃𝑅𝐸 ≅ ∆𝑃𝑅𝑈
5. Summarizing the
Lesson
How do you prove HA theorem?
1. Complete the Hypotenuse-Acute angle
Theorem
If the __________ and an ______ angle of
one__________ triangle are congruent to the
corresponding _________and an acute
___________ of another right triangle, then the
two triangles are ______.
2. Where can we see the wonders of Geometry in
our everyday living especially the things which
109
have same size and shape? Cite example. What
are the importance of having congruent
structures of buildings, bridges, etc.?
C. Assessment
Prove that the two triangles are congruent using
HyA Theorem
Given: ∠𝐾 𝑎𝑛𝑑 ∠𝑁 are right angles
̅̅̅
𝐼𝐷 bisects ∠𝐾𝐼𝑁
Prove:∆𝐾𝐼𝐷 ≅ ∆𝑁𝐼𝐷
Proof:
Statements
∠𝐾 𝑎𝑛𝑑 ∠𝑁 are right
angles
∠𝐾𝐼𝐷 ≅ ∠𝑁𝐼𝐷
Reasons
Definition of right
triangles
Given
Definition of
___________
̅̅̅ ≅ 𝐼𝐷
̅̅̅
𝐼𝐷
HA theorem
D. Agreement/Assignment:
Read about LL Congruence theorem and give
one example of proving this theorem.
V. REMARKS:
VI. REFLECTION:
A. No. of learners who earned
80% in the evaluation
B. No. of learners who require
additional activities for
remediation who scored
below 80%
C. Did the remedial lesson
work?
No. of learners who caught
up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked well?
Why did these work?
110
F. What difficulties did I
encounter which my
principal or supervisor can
help me solve?
G. What innovation or localized
materials did I use/discover
which I wish to share with
other teachers?
111
(M8GE-IIIh-1) - Proves Statements on Triangle Congruence
School:
Teacher:
Grade Level:
Learning
Area:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance Standards:
C. Learning Competencies:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
8
MATHEMATICS
THIRD
Week 8, Day 2
Quarter:
The learner demonstrates understanding of key concepts
of triangle congruence.
The learners shall be able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations.
The learner proves statements on triangle congruence
using HyL Congruence Theorem.
(M8GE-IIIh-1 )
Proving Right Triangle Congruence by HyL (HypotenuseLeg Congruence Theorem
pp. 392-393
pp. 361-362
Patterns and Practicalities by Gladys C. Nivera pp. 387388
http://m.wikipedia.org/wiki
Advanced Learners
IV. PROCEDURE
A. Preliminary Activity/
Motivation
Average Learners
“Be an Assessor!”
Maricel, Jhen, and Christian all own a triangular parcel
of lot at San Lorenzo, Tabaco City. Consider their lots
below. Without the help of the city assessor, which of the
following pieces of land have equal values?
Trivia: An assessor is a local government official who
determines the value of a property for local estate taxation
purposes
Maricel
112
Jhen
Christian
* How will you know that the two pieces of land have equal
values?
* Why is it that the triangular lot owned by Christian does
not have a value equal to the two others?
B. Presentation of the
Lesson
1. Problem Opener
Take a look at the image of the front area of Saint John
the Baptist church of Tabaco City which was founded on
1664.
Looking at it in aerial view, the area looks like this:



If we are to divide it in two triangles, say ∆𝐿𝐸𝑉 and
∆𝑉𝑂𝐿, how are we going to prove that the two
triangles are congruent?
Given that ∆𝐿𝐸𝑉 ≅ ∆𝑉𝑂𝐿, what statements can
you give to prove that they are congruent by HyA
Congruence Theorem?
Now, how about if we want to prove their
congruence by HyL (discussion of HyL). What
statements can you give? Let us fill in the table
below.
Given: ∠𝐿𝐸𝑉 𝑎𝑛𝑑 ∠𝑉𝑂𝐿 are right angles
̅̅̅̅
𝐿𝐸 ≅ ̅̅̅̅
𝑉𝑂
Statements
∆𝐿𝐸𝑉 ≅ ∆𝑉𝑂𝐿
113
Reasons
2. Group Activity
To understand better the Hyl Congruence Theorem, the
class will be grouped into 4. Each group will be given
different set of triangles to prove using the two-column
proof.
Group 1
Given: ̅̅̅̅
𝐶𝐴 ≅ ̅̅̅̅
𝐶𝐸
̅̅̅̅
𝐶𝑅 is perpendicular bisector of ̅̅̅̅
𝐴𝐸
Prove:∆𝐶𝑅𝐸 ≅ ∆𝐶𝑅𝐴
Statements
̅̅̅̅
𝐶𝐴 ≅ ̅̅̅̅
𝐶𝐸
Reasons
Given
Perpendicular lines form
right angles.
∆𝐶𝑅𝐸𝑎𝑛𝑑 ∆𝐶𝑅𝐴
are right triangles
Definition of perpendicular
bisector
∆𝐶𝑅𝐸 ≅ ∆𝐶𝑅𝐴
Group 2
̅̅̅̅ ≅ 𝐻𝑇
̅̅̅̅
Given: 𝐴𝐹
𝐼 is the midpoint of ̅̅̅̅
𝐴𝐻
∠𝐴𝐹𝐼 𝑎𝑛𝑑 ∠𝐻𝑇𝐼 are right angles
Prove: ∆𝐴𝐹𝐼 ≅ ∆𝐻𝑇𝐼
Statements
̅̅̅̅
𝐴𝐹 ≅ ̅̅̅̅
𝐻𝑇
̅̅̅
̅̅̅
𝐴𝐼 ≅ ̅𝐻𝐼
∠𝐴𝐹𝐼 and ∠𝐻𝑇𝐼
are right angles
∆𝐴𝐹𝐼 𝑎𝑛𝑑 ∆𝐻𝑇𝐼
are right triangles
Reasons
Given
Definition of ________
Definition of ________
HyL Congrue
Theorem
114
e
Group 3
̅̅̅̅ ≅ 𝑇𝐺
̅̅̅̅
Given: 𝐿𝐺
̅̅̅̅
𝐺 is the midpoint of 𝐼𝐻
∠𝐿𝐼𝐺 𝑎𝑛𝑑 ∠𝑇𝐻𝐺 are right angles
̅ ≅ ̅̅̅̅
Prove:𝐿𝐼
𝑇𝐻
Statements
̅̅̅̅
𝐿𝐺 ≅ ̅̅̅̅
𝑇𝐺
Reasons
Given
Definition of midpoint
∠𝐿𝐼𝐺 𝑎𝑛𝑑 ∠𝑇𝐻𝐺
are right angles
________ are right triangles.
∆𝐿𝐼𝐺 ≅ ∆𝑇𝐻𝐺
Definition of right triangle
CPCTC
Group 4
̅̅̅̅ ≅ 𝐸𝐹
̅̅̅̅
Given: 𝐵𝐶
̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐷𝐹
∠𝐴𝐵𝐶 𝑎𝑛𝑑 ∠𝐷𝐸𝐹 are right angles
Prove: ∠𝐶𝐵 ≅ ∠𝐷𝐹𝐸
St ements
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐸𝐹
Reasons
Given
∠𝐴𝐵𝐶 and ∠𝐷𝐸𝐹
are right angles
∆𝐴𝐵𝐶 and ∆𝐷𝐸𝐹
are right triangles
∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹
∠𝐶𝐵 ≅ ∠𝐷𝐹𝐸
Definition of ________
**A representative of each group will present their work.
115
3. Processing the
Answers
1. How did you find the activity?
2. From the name HyL, what parts of the triangles
are proven congruent?
3. Is HyL Congruence Theorem applicable to any
type of triangle? Why?
4. Did you use other stock knowledge to prove the
congruence? What are they?
4. Reinforcing the Skills
Given: ∠𝐴𝐻𝑀 𝑎𝑛𝑑 ∠𝐻𝐴𝑇 are right angles
̅̅̅̅̅ ≅ 𝑇𝐻
̅̅̅̅
𝑀𝐴
Prove: ∆𝐴𝐻𝑀 ≅ ∆𝐻𝐴𝑇
Statements
5. Summarizing the
Lesson
Reasons
* When are we going to use HyL Congruence Theorem?
What kind of triangles are we going to apply this theorem?
*How do we prove triangles using HyL congruence
theorem?
*What does HyL or HL congruence say?
Complete the Hypotenuse-Leg Theorem
If the __________ and ______ of a __________ triangle
are congruent to the _________ and ___________ leg of
another triangle, then the two triangles are ______.
C. Assessment
Prove that the two triangles are congruent:
I
L
E
116
F
To support the acacia tree during bad weather, wires
should be attached from the trunk of the tree to stakes in
the ground as shown above.
̅̅̅̅
Given: ̅̅̅
𝐼𝐸 ⟘ 𝐿𝐹
̅
𝐿𝐼 ≅ ̅̅̅
𝐹𝐼
Prove: ∠𝐼𝐿𝐸 ≅ ∠𝐼𝐹𝐸
Statements
̅̅̅
𝐼𝐸 ⟘ ̅̅̅̅
𝐿𝐹
Reasons
Given
̅̅̅
𝐼𝐸 ≅ ̅̅̅
𝐼𝐸
∠𝐼𝐸𝐿 𝑎𝑛𝑑 ∠𝐼𝐸𝐹 are right
angles
Definition of right triangles
∆𝐼𝐸𝐿 ≅ ∆𝐼𝐸𝐹
CPCTC
D. Agreement/
Assignment:
Read about HA(Hypotenuse-Angle) Congruence
Theorem
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for remediation
who scored below 80%
C. Did the remedial lesson
work?
No. of learners who
caught up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did I
use/discover which I wish
to share with other
teachers?
117
(M8GE-IIIh-1) - Proving Statements on Triangle Congruence
School:
Teacher:
Time and Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance
Standards:
C. Learning
Competencies:
II. CONTENT:
III. LEARNING
RESOURCES:
A. References
1. Teacher’s Guide
Pages
2. Learner’s Guide
Pages
3. Textbook Pages
4. Additional Material
from Learning
Resource Material
B. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/Prelimi
nary Activity
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 8, Day 3
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner proves that statements on right triangle
congruence by LL (Leg – Leg) Congruence Theorem.
(M8GE-IIIh-1)
Proving Statements on Right Triangle Congruence using
Leg – Leg (LL) Congruence Theorem)
pp. 392-393
pp. 361-362
https://googleimagesbicolregionmap.com
Advanced Learners
Average Learners
“HUMAN TEXT TWIST”
A number of students will be selected and each of them
will be given a letter. They must arrange themselves to form
a word or term.
1.
2.
3.
4.
5.
6.
TIGRH
RNEGTLIA
EGL
PHTNSEUEOY
CTEAU
NLEAG
To what particular triangle congruence can we correlate
each of the words formed?
118
B. Presentation of the
Lesson
1. Problem
Opener/ Group
Activity
Given: ̅̅̅̅
OY ⊥ ̅̅̅̅
BY and ̅̅̅̅
EN ⊥ ̅̅̅̅
SN
̅̅̅̅ ≅ ̅̅̅̅
BY
SN and ̅̅̅̅
OY ≅ ̅̅̅̅
EN
Mark the figures below to show the given information then
use a two-column proof to prove that ∆BOY ≅ ∆SEN.
STATEMENT
̅̅̅̅
𝐁𝐘 ≅ ̅̅̅̅
𝐒𝐍
REASON
̅̅̅̅
𝐎𝐘 ≅ ̅̅̅̅
𝐄𝐍
̅̅̅̅
𝐎𝐘 ⊥ ̅̅̅̅
𝐁𝐘 and ̅̅̅̅
𝐄𝐍 ⊥ ̅̅̅̅
𝐒𝐍
∠𝐘 𝐚𝐧𝐝 ∠𝐍 are right
angles
∠𝐘 ≅ ∠𝐍
∆𝐁𝐎𝐘 ≅ ∆𝐒𝐄𝐍
2. Processing the
answer (Analysis)
1. What kind of triangles did you prove congruent?
2. What side/s and angle/s of ∆BOY and ∆SEN are given
congruent?
3. What congruence postulate did you use to prove that
the two triangles are congruent?
4. What do you call the perpendicular sides of the right
triangle?
In right triangles, SAS (Side – Angle – Side) Congruence
Theorem is referred to as the LL (Leg – Leg)
Congruence Theorem.
The proof you have shown is the proof of the LL
Congruence Theorem.
Complete the statement:
If the ___________ of one right triangle are congruent
to the corresponding ______________ of another right
triangle, then the triangles are _________________.
119
3. Reinforcing the
Skills (Application)
GROUP 1
Complete the two-column proof below to show that
̅̅̅̅
̅̅̅̅
PE ≅ DO. Refer to the figure below:
Given: ̅̅̅̅
ET ≅ ̅̅̅̅
OG
̅̅̅̅ ≅ DG
̅̅̅̅
PT
∠T and ∠G are right angles
(Mark the illustration based from the given information)
Prove: ̅̅̅̅
PE ≅ ̅̅̅̅
DO
STATEMENT
̅̅̅̅
ET ≅ ̅̅̅̅
OG
REASON
Given
∠T and ∠G are right
angles
Definition of Right
Triangles
∆PET ≅ ∆DOG
̅̅̅̅
PE ≅ ̅̅̅̅
DO
GROUP 2
This KIOSK/GAZEBO is in the middle part of
Tabaco City Park. It is said that its roof is perfectly designed
with six congruent triangles. If we draw a perpendicular line
from the top most vertex to the opposite side of one of the
triangles, lets prove that ∠𝐁 𝐚𝐧𝐝 ∠𝐒 are congruent using the
other given information also.
120
̅̅̅̅ ⊥ BS
̅̅̅̅
Given: ET
̅̅̅
T is midpoint of ̅BS
(Mark the illustration based from the given information)
STATEMENT
̅̅̅̅
̅̅̅
ET ⊥ ̅BS
m∠BTE = 90
m∠STE = 90
REASON
Given
Definition of Right
Triangles
̅̅̅
T is midpoint of ̅BS
Given
̅̅̅̅
BT ≅ ̅̅̅
ST
Reflexive Property
∆BET ≅ ∆SET
∠B and ∠S
GROUP 3
Jen and Chen are both natives of Albay. They are
planning to have a vacation in at most two different
provinces of Bicol. Jen wants to include Camarines Sur to
their travel goals but Chen had already explored Cam Sur
and Chen wants to visit Sorsogon but Jen had been there
already. So they have just decided to go separately and just
meet at Catanduanes which will be the second itinerary of
each of them. And from there, they will go back to Albay
together.
Looking at the Map of Region V, they noticed that
their itinerary forms two triangles. Suppose ̅̅̅̅
𝑂𝐷 is a
̅̅̅̅, prove that 𝐺𝑂
̅̅̅̅ ≅ 𝐿𝑂
̅̅̅̅. (Mark the
perpendicular bisector of 𝐺𝐿
triangles to show the given information)
121
STATEMENT
̅̅̅̅̅
𝑶𝑫 is a perpendicular
̅̅̅̅
bisector of 𝑮𝑳
̅̅̅̅
̅̅̅̅
𝑮𝑫 ≅ 𝑳𝑫
m∠𝐎𝐃𝐋 =
𝟗𝟎 𝐚𝐧𝐝 𝐦∠𝐎𝐃𝐆 = 𝟗𝟎
REASON
Given
Definition of Right
Triangles
Reflexive Property
∆𝐆𝐎𝐃 ≅ ∆𝐋𝐎𝐃
̅̅̅̅
𝑮𝑶 ≅ ̅̅̅̅
𝑳𝑶
GROUP 4
To complete the lyrics of the chorus of Tabaco City
Hymn below, supply the reasons for the given statements
in the two-column proof. Each reason has a corresponding
missing lyric of the song.
CHORUS
Go! Tabaco City
Grow Tabaco City
1. _____ Tabaqueños ______!
Let 2. __________________
Let 3. _____________________
Pledge to 4. ___________________
Let 5. ____________________
In surpassing All.
(Repeat Chorus)
Complete the Two Column Proof by choosing the
correct reasons in the table below with their corresponding
lyrics.
CORRESPONDING
REASON
LYRICS
Definition of Right Angles
Grow, Go
CPCTC
One vision lead us on
Definition of
Perpendicular Segments
Go, Grow
LL Congruence Postulate
Work with love
Definition of Right
Triangles
Us heed the call
Reflexive Property
Us raise our hands
now
122
̅̅̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅̅
̅̅̅̅̅
Given: 𝐷𝑀
𝐸𝐴 are ⊥ 𝑡𝑜 𝑀𝐴
̅̅̅̅̅̅ ≅ 𝐸𝐴
̅̅̅̅̅
𝐷𝑀
(Mark the illustration based from the given information)
Prove: ∠𝐷 ≅ ∠𝐸
STATEMENT
̅̅̅̅̅̅
𝐷𝑀 𝑎𝑛𝑑 ̅̅̅̅̅
𝐸𝐴 are
̅̅̅̅̅
⊥ 𝑡𝑜 𝑀𝐴
∠𝐷𝑀𝐴 =900 and
∠𝐸𝐴𝑀=900
∆DMA and ∆EAM
are right triangles
̅̅̅̅̅ ≅ 𝐴𝑀
̅̅̅̅̅
𝑀𝐴
̅̅̅̅̅̅ ≅ 𝐸𝐴
̅̅̅̅̅
𝐷𝑀
REASON
CORRESPONDING
LYRICS
Given
*********************
Given
*********************
∆DMA ≅ ∆EAM
4. Summarizing the
Lesson
∠𝐷 ≅ ∠𝐸
1. What does the LL (Leg – Leg) Congruence Theorem
state?
2. What are the things to consider in proving statements on
Right Triangle Congruence using the LL Congruence
Theorem?
C. Assessment
∠𝐷𝑂𝐺 = 900
̅̅̅̅̅ 𝒂𝒏𝒅 𝐆𝐓
̅̅̅̅ bisect each other at point O
𝐇𝐃
(Mark the illustration based from the given information)
Given:
Prove:
̅̅̅̅ 𝐓𝐇
̅̅̅̅
𝐆𝐃
123
STATEMENT
̅̅̅̅
̅̅̅̅
HD 𝑎𝑛𝑑 GT bisect each other at
point O
̅̅̅̅
HO ≅ ̅̅̅̅
DO and ̅̅̅̅
GO ≅ ̅̅̅̅
TO
REASON
∠𝐷𝑂𝐺 = 900
∠𝐷𝑂𝐺 ≅ ∠𝐻𝑂𝑇
Given
Given
∆HOT and ∆DOG are right triangles
∆HOT ≅ ∆DOG
̅̅̅̅ 𝑎𝑛𝑑 TH
̅̅̅̅
GD
D. Agreement:
Read about LA (Leg – Acute Angle) Congruence
Theorem.
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for remediation
who scored below 80%
C. Did the remedial
lesson work?
No. of learners who
caught up with the lesson
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized materials did I
use/discover which I
wish to share with other
teachers?
124
(M8GE-IIIh-1) - Proving Statements on Triangle Congruence
School:
Teacher:
Time and
Date:
I. OBJECTIVES:
A. Content Standards:
B. Performance Standards:
C. Learning Competency:
II. CONTENT:
III. LEARNING RESOURCES:
E. References
5. Teacher’s Guide
Pages
6. Learner’s Guide Pages
7. Textbook Pages
8. Additional Material
from Learning
Resource Material
F. Other Learning
Resources
IV. PROCEDURES:
A. Motivation/Preliminary
Activity
Grade Level:
Learning Area:
Quarter:
8
MATHEMATICS
THIRD
Week 8, Day 4
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
The learner proves statements on right triangle
congruence by LA (Leg – Acute Angle) Congruence
Theorem. (M8GE-IIIh-1)
Proving Statements on Right Triangle Congruence using
Leg – Acute Angle (LA) Congruence Postulate)
pp. 392-393
pp. 361-364
Advanced Learners
Average Learners
Recall of the First Three Right Triangle Congruence
Theorems (HyL, HyA, LL)
“Jumbled Words and Symbols”
The class will be divided into two groups where in each
group there will be 10 student members. Each student will
be given a word/phrase/symbol. A question will be raised
to the class and the student holding the
words/phrases/symbols needed to form the answer will go
in front and arrange themselves to form the answer. The
words/phrases/symbols included are;
1. Leg
2. Congruence
3. Hypotenuse
4. Acute angle
5. Theorem
6. Of one Right ∆
7. then the ∆𝑠 are ≅
8. If the legs
9. Legs of another Right ∆
10. Are ≅ to the
125
1. What right triangle congruence theorem states that if the
hypotenuse and an acute angle of one right triangle are
congruent to the corresponding hypotenuse and an acute
angle of another right triangle, then the triangles are
congruent?
2. What does the LL (Leg – Leg) Congruence Theorem
states?
3. What right triangle congruence theorem states that if the
hypotenuse and leg of one right triangle are congruent to
the corresponding hypotenuse and a leg of another right
triangle, then the triangles are congruent?
B. Presentation of the
Lesson
1. Problem Opener
̅̅̅̅̅ ⊥ ̅̅̅̅̅
1. Given: TW
OW
̅̅̅̅
̅̅̅
ML ⊥ SL
̅̅̅̅̅
TW ≅ ̅̅̅̅
ML
∠𝐓 ≅ ∠𝐌
Mark the figures below to show the given information then
use a two-column proof to prove that ∆OTW ≅ ∆SML.
STATEMENT
̅̅̅̅̅
TW ⊥ ̅̅̅̅̅
OW
REASON
̅̅̅̅
ML ⊥ ̅̅̅
SL
∠𝐖 𝐚𝐧𝐝 ∠𝐋 are right
angles
∠𝐖 ≅ ∠𝐋
∠𝐓 ≅ ∠𝐌
∆𝐎𝐓𝐖 ≅ ∆𝐒𝐌𝐋
126
ASA Congruence
Postulate
Group Activity
GROUPS 1 & 2
This is a picture of a bus stop in Tabaco City
̅̅̅̅ ⊥ ̅̅̅̅
Given: LE
BU
̅̅̅̅ is an angle bisector of ∠𝐵𝐿𝑈
LE
Prove: ∠B and ∠S
(Mark the illustration based from the given information)
STATEMENT
̅̅̅̅ ⊥ ̅̅̅̅
LE
BU
m∠BEL = 90
m∠UEL = 90
REASON
Given
Definition of Right
Triangles
̅̅̅̅ is an angle
LE
bisector of ∠𝐵𝐿𝑈
∠BLE ≅ ∠ULE
Given
Reflexive Property
∆BEL ≅ ∆UEL
∠B ≅ ∠S
GROUP 3 & 4
To complete the lyrics of the Bicol Regional March
below, supply the reasons for the given statements in the
two-column proof. Each reason has corresponding missing
lyrics of the song.
CODA
Bicolandia! Bicolandia!
The 1. _________________________
2. ____________ warriors, 3. _______________
To 4. _________________, 5. ___________
Repeat Coda
To 4. _____________, 5. _________________
127
Complete the Two Column Proof by choosing the correct
reasons in the table below with their corresponding lyrics.
CORRESPONDING
REASON
LYRICS
Definition of Right Angles
Oragons’ Home
CPCTC
We give in
Definition of Perpendicular
Segments
Home of the Oragons
LA Congruence Theorem
Truth and dignity
Definition of Right Triangles
Fearless
Reflexive Property
Bold yet plain
Given: ̅̅̅̅̅̅
𝐷𝑀 𝑎𝑛𝑑 ̅̅̅̅̅
𝐸𝐴 are ⊥ 𝑡𝑜 ̅̅̅̅̅
𝑀𝐴
∠𝐷 ≅ ∠E
(Mark the illustration based from the given information)
Prove: ̅̅̅̅
ME ≅ ̅̅̅̅
AD
STATEMENT
̅̅̅̅̅̅
𝐷𝑀 𝑎𝑛𝑑 ̅̅̅̅̅
𝐸𝐴 are
̅̅̅̅̅
⊥ 𝑡𝑜 𝑀𝐴
∠𝐷𝑀𝐴 =900 and
∠𝐸𝐴𝑀=900
∆DMA and ∆EAM
are right triangles
̅̅̅̅̅
𝑀𝐴 ≅ ̅̅̅̅̅
𝐴𝑀
∠𝐷 ≅ ∠𝐸
∆DMA ≅ ∆EAM
̅̅̅̅ ≅ AD
̅̅̅̅
ME
128
REASON
CORRESPONDING
LYRICS
Giv n
********************
Given
*********************
2. Processing the
answer (Analysis)
1. What kind of triangles did you prove congruent?
2. What are the reasons you’ve used for the statements?
3. What congruence postulate did you use to prove that
the two triangles are congruent?
4. What parts of the triangle must be first proven
congruent before you can conclude that two right
triangles are congruent by LA Congruence Postulate?
Complete the statement:
If the ___________ of one right triangle are congruent
to the corresponding ______________ of another right
triangle, then the triangles are _________________.
3. Reinforcing the skills
̅̅̅̅ ⊥ OA
̅̅̅̅ ⊥ ̅̅̅̅
Given: TO
AY
̅̅̅̅
̅̅̅̅
TO ≅ YA
(Mark the illustration based from the given information)
Prove:
̅̅̅̅ ≅ 𝐘𝐃
̅̅̅̅
𝐓𝐃
STATEMENT
REASON
̅̅̅̅
TO ⊥ ̅̅̅̅
OA ⊥ ̅̅̅̅
AY
Given
∠𝑂 = 900 and ∠𝑨 = 𝟗𝟎°
∆TOD AND ∆YAD are right triangles
∠𝑇𝐷𝑂 ≅ ∠𝑌𝐷𝐴
̅̅̅̅ ≅ ̅̅̅̅
TO
YA
∆TOD ≅ ∆YAD
̅̅̅̅
𝐓𝐃 ≅ ̅̅̅̅
𝐘𝐃
4. Summarizing the
Lesson (Abstraction)
1. What does the LA (Leg – Acute Angle) Congruence
Theorem state?
2. What are the things to consider in proving statements
on Right Triangle Congruence using the LA
Congruence Theorem?
129
C. Evaluation
Given: ̅̅̅̅
OE ⊥ ̅̅̅̅
DV
̅̅̅̅
OE is an angle bisector of ∠𝐷𝑂𝑉
̅̅̅̅ ≅ 𝐘𝐃
̅̅̅̅
Prove: 𝐓𝐃
(Mark the illustration based from the given information)
ST TEMENT
̅̅̅̅ ⊥ ̅̅̅̅
OE
DV
m∠OED = 90
m∠VEO = 90
RE SON
Given
Definition of Right
Triangles
̅̅̅̅ is an angle
OE
bisector of ∠𝐵𝐿𝑈
∠VOE ≅ ∠DOE
Given
Reflexive Property
∆BEL ≅ ∆UEL
̅̅̅̅ ≅ 𝐘𝐃
̅̅̅̅
𝐓𝐃
D. Agreement:
1. Define Perpendicular Lines and Angle Bisector
2. How do we apply or use triangle congruence in
constructing perpendicular lines and angle bisectors?
V. REMARKS:
VI. REFLECTION:
A. No. of learners who
earned 80% in the
evaluation
B. No. of learners who
require additional
activities for remediation
who scored below 80%
C. Did the remedial lesson
work?
No. of learners who
caught up with the lesson
D. No. of learners who
continue to require
remediation.
130
E. Which of my teaching
strategies worked well?
Why did these work?
F. What difficulties did I
encounter which my
principal or supervisor can
help me solve?
G. What innovation or
localized materials did I
use/discover which I wish
to share with other
teachers?
131
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisector
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
8
MATHEMATICS
THIRD
Week 9, Day 1
The learner demonstrates understanding of key concepts of
axiomatic developments of Geometry.
The learner is able to communicate mathematical thinking with
coherence and clarity in formulating, investigating, analyzing
and solving problems.
The learner is able to construct congruent triangles by SSS
(M8GE-IIIi-j-1)
Applying triangle
triangles)
congruence
(Constructing
III.
congruent
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning
Resources
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary
ACTIVITY 1
Activities/
Study the markings. Indicate whether or not the triangles in
Motivation
each item are congruent. If so, write a congruence statement
and determine the postulate used to prove the congruency.
1.
2.
4.
132
3.
B. Presentation of
the Lesson
1. Problem Opener
2. Group Activity
 Consider the pairs of congruent triangles in Activity 1:
a. ∆ ADB ≅ ∆CBD, by SSS Congruence Postulate
b. ∆ AMT ≅ ∆HMT, by SAS Congruence Postulate
c. ∆ BES ≅ ∆MEN, by ASA Congruence Postulate
 Is it possible to construct another triangle that is
congruent to ∆ ADB 𝑎𝑛𝑑 ∆CBD? If yes, how?
 What geometry tools do you use in constructing figures
accurately?
ACTIVITY 2 (by group)
(Each group will perform the task using a compass and a
straight edge on a colored paper. They will be instructed to cut
out the triangles they made and show them in class)
TASK: Given sides AB, BC and AC, construct ∆ABC.
A
B
A
STEPS:
1. Draw a reference line
and mark a starting
point A.
2. Construct one of the
sides, say AC, on the
reference line. To do
this, set the compass
to radius AC and draw
an arc with center A
such that it intersects
the reference
line.(Label the
intersection (point) C.
3. Set the compass to
radius AB and draw an
arc with center A,
above the reference
line
133
B
C
C
Expected Output
4. Set the compass to
radius BC, place its
point to point C of
constructed segment
AC and draw an arc
so it intersects the
previous arc.
5. Label the arcs’ point of
intersection (point) B.
6. Connect the points to
draw side AB and BC.
3. Processing the
Answer
 What can you say about your constructed triangles? Match
their vertices and see whether they coincide.
 Would you agree that all triangles constructed using these
given segments are congruent? Why or why not?
 What triangle congruence postulate is illustrated in this
task?
 How important is it to use appropriate tools in constructing
congruent triangles?
4. Reinforcing the
Skills
Construct a triangle whose side lengths are 6cm, 7cm and
4cm.
5. Summarizing the
Lesson
How do you construct congruent triangles by SSS?
C. Assessment
Construct ∆KEY congruent to Construct a triangle whose
∆ PAD whose side lengths are side lengths are 7cm, 10cm
the following:
and 5 cm.
PA= 14.5cm, AD= 18.6cm
and
PD= 13.6cm
134
D. Agreement/
Assignment
Suppose the ratio of the side lengths of ∆ABC is 2:2:3 and
its perimeter is 35cm, construct a triangle congruent to
∆ABC.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized material/s did
I use/discover which I
wish to share with
other teachers?
135
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisector
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
8
MATHEMATICS
THIRD
Week 9, Day 2
The learner demonstrates understanding of key concepts of
axiomatic developments of Geometry.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing and solving problems.
The learner is able to construct congruent triangles by SAS
and ASA
(M8GE-IIIi-j-1)
Applying triangle
triangles)
congruence
Constructing
congruent
III.
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s Materials
pages
3. Textbook pages
4. Additional
Materials
from
Learning Resource
(LR) portal
B. Other
Learning
Resources
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary Activities/
ACTIVITY 1 (Recall on constructing congruent segments
Motivation
and angles) Construct a segment and an angle congruent to
̅̅̅̅
𝐴𝐵 and ∠𝐴, respectively. The following steps are given.
Construct: ̅̅̅̅
𝐶𝐷 congruent to Construct: ∠𝑊 congruent
̅̅̅̅
to ∠𝐴
𝐴𝐵
136
STEPS:
STEPS:
1. Draw a reference line and 1. Draw a ray with
mark a starting point C.
endpoint W.
2. Set the compass to radius 2. Draw a circular arc with
AB and draw an arc with
center at A and cutting
center C crossing the
the sides of ∠A at point
reference line.
B and C, respectively.
3. Label the intersection as 3. Draw a similar arc
point D.
using the center W and
̅̅̅̅
̅̅̅̅
radius AB, intersecting
4. 𝐴𝐵 ≅ 𝐶𝐷
the ray at X.
4. Set
the
compass
opening
to
length
radius BC.
5. Using X as center and
BC as radius, draw an
arc intersecting the first
arc at Y
6. Draw ray ̅̅̅̅̅
𝑊𝑌 to
complete
∠𝑊 congruent to ∠𝐴
B. Presentation of the
Lesson
1. Problem Opener
ACTIVITY 2: Consider the given sets of segments and
angles.
 How many segments and angles are given in SET
A? How about in SET B?
 Name the segments and angles given in SET A and
SET B.
2. Group Activity
(By Group) Using the segments and angles given in
activity 2, perform the following tasks:
137
(GROUPS 1 & 2)
(GROUPS 3&4)
TASK: Construct ∆ABC
given ∠A and segments b
and c.
TASK: Construct ∆LMN
given ∠M, ∠L and segment
n
1. Draw a reference line
and mark a starting point
A.
2. To construct/copy
segment c, set the
compass to radius c.
Keeping this radius,
place the compass point
on A and draw a small
arc crossing the
reference line. (Label B)
3. To construct/copy ∠A,
place the compass point
at the vertex of the given
angle. Draw an arc
crossing both sides of
the angle. (Keep this
compass opening)
4. Without changing the
compass opening, place
the compass point at A
on the reference line and
draw an arc large
enough and crossing the
reference line.
5. Return to the given
angle. Using the
compass, measure the
opening across the arc.
6. Without changing the
compass opening, place
the compass point at the
intersection of the bigger
arc and the reference
line. Draw another arc
crossing the larger arc.
7. Draw a ray from point A
through the point of
1. Draw a reference line
and mark a starting
point M.
2. To construct/ copy
∠M, place the compass
point at the vertex M of
the given angle. Draw
an arc crossing both
sides of angle M.
3. Without changing the
compass opening, place
the compass point at M
on the reference line
and draw an arc large
enough and crossing
the reference line.
4. Return to the given
angle M and measure
the opening across the
arc. (Keep this compass
opening)
5. Without changing the
compass opening, place
the compass point at the
intersection of the larger
arc and the reference
line. Draw an arc
crossing the larger arc.
6. To construct/ copy
segment n, set the
compass to radius n.
Place the compass point
at M on the reference
138
intersection of the arcs
above the reference line.
8. To construct/ copy
segment b, place the
compass point at B on
the reference line and
draw an arc crossing the
ray (label c). Connect the
points A, B, and C to
draw ∆ABC.
3. Processing the
Answer
4. Reinforcing the Skills
5.Summarizing the
Lesson
C. Assessment
line and draw a small
arc crossing the
reference line. (Label L)
7. (Copy angle L)Repeat
directions above for
copying ∠L at point L.
8. Label the point of
intersection of two angle
rays as N. Draw ∆LMN.
 (Let the students show their constructed triangles in
class)
 Compare triangles constructed by Groups 1& 2.
Also, triangles made by groups 3 &4.
 Is group 1’s triangle congruent to that of group 2’s?
 How about the triangles made by Group 3&4, are
they congruent?
 (Let the students cut out the triangles to verify)
 Would you agree that all triangles constructed out of
the given segment/s and angle/s in SET A are
congruent? How about in SET B?
 Notice that in the first task, the construction of the
triangle follows the ‘COPY SEGMENT- COPY
ANGLE- COPY SEGMENT’ order. What triangle
congruence postulate is being illustrated in this
task?
 Notice that in the 2nd task, however, the construction
of the triangle follows the ‘COPY ANGLE-COPY
SEGMENT-COPY ANGLE’ order. What triangle
congruence postulate is being illustrated in this
task?
1. Construct ∆ HAT congruent to ∆POD given
∠0 = 40˚ , side p =7cm and side d =6cm
2. Construct ∆ BEG congruent to ∆ROS given
∠0 = 37 ˚ , ∠𝑅 = 45˚ and side 𝑠 = 8𝑐𝑚
How do you construct congruent triangles by SAS and ASA?
Construct ∆QRS congruent
to ∆ABC whose angle
measure/s and segment
length/s are given.
1. ∠𝑨 = 𝟏𝟑𝟓˚; b=5cm;
c=6cm
2. ∠A =105˚ : ∠C =40˚ ;
b= 4.5 cm
139
Construct a triangle
congruent to the following
triangles.
1.
2.
D. Agreement/
Assignment
Construct ∆BEC congruent to ∆SPA given ∠S= 131˚,
∠P=16˚ and a= 8.5cm.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did
the
remedial
lessons work? No. of
learners who have
caught up with the
lesson.
D. No. of learners who
continue to require
remediation.
E. Which of my teaching
strategies
worked
well? Why did it work?
F. What difficulties did I
encounter which my
principal or supervisor
can help me solve?
G. What innovation or
localized
material/s
did I use/discover
which I wish to share
with other teachers?
140
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisector
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide
pages
2. Learner’s
Materials
pages
3. Textbook pages
4. Additional Materials
from
Learning
Resource (LR) portal
B. Other
Learning
Resources
IV.
PROCEDURE
A. Preliminary Activities/
Motivation
GRADE 8
MATHEMATICS
THIRD
Week 9, Day 3
The learner demonstrates understanding of key
concepts of axiomatic developments of Geometry.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing and solving problems.
The learner is able to construct congruent triangles by
SSS, SAS and ASA
(WORKSHEET) (M8GE-IIIi-j-1)
Applying triangle congruence
Compass, straightedge or ruler, construction paper,
worksheets
ADVANCE LEARNERS
AVERAGE LEARNERS
A. Checking of materials to be used in the task
B. Recall on how to construct congruent
triangles.
141
WORKSHEET: ________
CONSTRUCTING CONGRUENT TRIANGLES BY SSS, SAS AND
ASA
Name: _________________________ Date_________ Gr. &Sec_______
A. Construct the following triangles by SSS.
1. 4cm, 9cm and 10 cm
2. 5cm, 8cm, 12cm
3. 4cm, 7cm, 8.5cm
B. Construct the triangles by SAS
1. 5cm, 8cm, 127˚
2. 4cm, 5cm, 30˚
C. Construct the triangle by ASA.
1. 7cm, 35˚, 59˚
142
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisector
School
Teacher
Grade Level
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
GRADE 8
MATHEMATICS
THIRD
Week 9, Day 4
The learner demonstrates understanding of key
concepts of axiomatic developments of Geometry.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing and solving problems.
The learner is able to construct an angle bisector and
perpendicular lines
(M8GE-IIIi-j-1)
Applying triangle congruence in Constructing an Angle
Bisector and Perpendicular Lines
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning Compass, straightedge or ruler, visual aids,
Resources
construction paper
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary
Activities/ Motivation
GUESS WHAT?
ACTIVITY 1. Fill out the boxes with the letters of the
term being described in each item.
This is a line/ray/segment that divides an angle into
two congruent angles.
These are lines that intersect at right angles.
143
B. Presentation of the
Lesson
2. Problem Opener
ACTIVITY 2:
I. Consider the following tasks:
A) Construct: The bisector of ∠A
B.) Construct: Line through P perpendicular to l
C.) Construct: Line through P perpendicular to l
2. Group Activity
(BY GROUP)
II. Follow the steps to perform each task in Activity 2 ,I.
A. GIVEN: ∠A
CONSTRUCT: Bisector of ∠A
STEPS:
1. Locate points B and C one on each side of ∠A so
that AB=AC. This can be done by drawing an arc
with center at A.
2. Using C as the center and any radius r which is more
than half of BC, draw an arc in the interior of A.
3. Then using B as the center, construct an arc with
the same radius r and intersecting the arc in the
preceding step at point X.
⃗⃗⃗⃗⃗ is the bisector of ∠BAC.
4. 𝐴𝑋
B. GIVEN : Line
l and point P on l
CONSTRUCT: Line through P perpendicular to l
144
STEPS:
1. Place the compass point on P, and draw arcs to cut
the given line on both sides of point P.
2. Place the compass point where the arc intersect the
line on one side and draw a small arc above the line.
3. Without changing the compass opening, place the
compass point at the intersection of the given line
and the arc on the OTHER side and draw another
arc above the given line (the two arcs above the
number line must intersect)
4. Using a straight edge, connect the intersection of
the two arcs to P.
C. GIVEN : Line
l and point P
NOT on
l
CONSTRUCT: Line through P perpendicular to l
STEPS:
1. Place the compass point on P and draw arcs
crossing line l on both sides of P.
2. Place the compass point where the arc
intersects the line on one side and draw a
small arc below the given line.
3. Without changing the compass opening, place
the compass point at the intersection of the
given line and the arc on the other side and
draw another arc below the given line in such
a way that it intersects the arc in the preceding
step.
4. Using a straight edge, connect the intersection
of these two arcs to P.
3. Processing the
Answer
4. Reinforcing the Skills
 Refer to task A, if ̅̅̅̅
𝐴𝑋 is the bisector of ∠BAC,
how is ∠BAX related to ∠CAX?
 Is there any way to verify that ∠BAX ≅ ∠CAX
? If there is, how?
 Refer to task B &C. using a protractor, verify
whether the angles formed by the lines are
90˚-angles.
A. Construct the bisector of ∠R
145
B. Construct a line perpendicular to line
point
m through
5.Summarizing the
Lesson
How do you construct an angle bisector and
perpendicular lines?
C. Assessment
1. Construct bisector of
∠C.
2. Construct a line
perpendicular to line g
passing through point H.
D. Agreement/
Assignment
1. Construct a line
perpendicular to line b
through P.
2. Construct bisector of
∠D.
Construct the altitudes of the given triangle.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
146
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners who
continue to require
remediation.
E. Which
of
my
teaching strategies
worked well? Why
did it work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to
share with other
teachers?
147
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisectors
School
Grade Level GRADE 8
Teacher
Learning
Area
Quarter
Time & Date
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
III.
LEARNING
RESOURCES
MATHEMATICS
THIRD
Week 10, Day 1
The learner demonstrates understanding of key
concepts of axiomatic structure of geometry and
triangle congruence.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life
problems involving congruent triangles using
appropriate and accurate representations.
Applies
triangle
congruence
to
construct
perpendicular lines and angle bisectors.
(M8GE-IIIi-j-1)
Applying
triangle
congruence
(Constructing
Perpendicular Lines)
Materials: Board Compass, Student Compass,
Straight Edge, Chalk and Chalkboard, construction
papers, marker, manila paper
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
Next Century Math pages 440 and 445-447
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/anderson/geometry/Resources
assignment14/assn14.html
ADVANCED LEARNERS
AVERAGE LEARNERS
IV.
PROCEDURE
A. Preliminary
ACTIVITY 1
Activities/ Motivation
In the figure, identify pairs of lines that are
perpendicular. (When are lines considered
perpendicular?
148
B. Presentation of
the Lesson
1. Problem
Opener
2. Group Activity
ACTIVITY 2
In the figure, perform the following:
1. Draw a circle whose center is
at B passing through A.
2. Draw a circle whose center is
at C passing through A.
3. Name the other intersection of
the two circles as D. Connect points A and D using
a line.
4. Connect points B and D and points C and D to
form ∆𝐴𝐵𝐶.
Answer the following questions:
1. Which two lines are perpendicular?
2. What can you say about ∆𝐴𝐵𝐶 and ∆𝐵𝐶𝐷? Show
or prove your observation. (Student may prove
that the triangles are congruent or show their
congruence using ruler and protractor.)
ACTIVITY 3 “MANIPU-TRIANGLES”
(Can we use congruent triangles to construct
perpendicular lines?)
For Advanced Learners
1. Cut out a pair of congruent right triangles and use
these triangles to construct a line perpendicular to
line 𝑚.
2. Cut out a pair of congruent acute triangles and
use these triangles to construct a line perpendicular
to line 𝑛.
3. Cut out a pair of congruent obtuse triangles and
use these triangles to construct a line perpendicular
to line 𝑝.
For Average Learners
1. Use the two congruent
right triangles to
construct a line
perpendicular to line 𝑚.
2. Use two congruent
acute triangles to
construct a line
perpendicular to line 𝑛.
3. Use two congruent
obtuse triangles to
construct a line
perpendicular to line 𝑝.
149
(Expected Answers) Show that they are
perpendicular using the protractor.
1.
2.
3. Processing the
Answer
4. Reinforcing the
Skills
5.Summarizing
the Lesson
C. Assessment
D. Agreement/
Assignment
3.
Describe the position of the two congruent triangles
in order to construct perpendicular lines.
● The triangles should not overlap or there should
be no common interior points.
● One of the pairs of congruent sides should
coincide.
● The remaining corresponding sides should share
the same endpoint.
̅̅̅̅. Name the
Construct a line perpendicular to 𝐴𝐵
congruent triangles formed when constructing the
perpendicular lines.
How can congruent triangles be used in
constructing perpendicular lines?
̅̅̅̅
In ∆𝑋𝑄𝑃, construct a line perpendicular to 𝑋𝑄
passing through the vertex 𝑃. Name the triangle
formed that is congruent to ∆𝑋𝑄𝑃.
Construct the perpendicular line on each side of the
triangle passing through a respective vertex and
identify all the congruent triangles formed.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
150
B.
C.
D.
E.
F.
G.
formative
assessment
No. of learners who
require additional
activities
for
remediation.
Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
No. of learners who
continue to require
remediation.
Which
of
my
teaching strategies
worked well? Why
did it work?
What difficulties did
I encounter which
my principal or
supervisor
can
help me solve?
What innovation or
localized material/s
did I use/discover
which I wish to
share with other
teachers?
151
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisectors
School
Grade Level
GRADE 8
Teacher
Learning
Area
Quarter
MATHEMATICS
Time & Date
I.
OBJECTIVES
A. Content Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
THIRD
Week 10, Day 2
The learner demonstrates understanding of key
concepts of axiomatic structure of geometry and
triangle congruence.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations.
Applies triangle congruence to construct perpendicular
lines and angle bisectors.
Applying
triangle
Perpendicular Lines)
III.
congruence
(Constructing
LEARNING
RESOURCES
A. References
1. Teacher’s Guide
pages
2. Learner’s
Materials pages
3. Textbook pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning http://jwilson.coe.uga.edu/emt668/emt668.folders.f97/anderson/geometry/Resources
assignment14/assn14.html
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary
ACTIVITY 1
ACTIVITY 1
Activities/
The two triangles are In the figure, all the
Motivation
congruent. Use the other triangles are congruent.
triangle to draw all the TASK: Draw three
altitudes of ∆𝐴𝐵𝐶.
altitudes of ∆𝐴𝐵𝐶.
152
B. Presentation of the
Lesson
1. Problem Opener
2. Group Activity
3. Processing the
Answer
4. Reinforcing the
Skills
ACTIVITY 2
Given: ∆𝐶𝐻𝐴 is an equilateral triangle.
1. Cut out a triangle congruent to ∆𝐶𝐻𝐴.
2. Use the cut out to construct the altitude of ∆𝐶𝐻𝐴
̅̅̅̅. Name the altitude as 𝐻𝑅
̅̅̅̅ .
from vertex 𝐻 to 𝐶𝐴
Answer or perform the following:
1. How is ̅̅̅̅
𝐻𝑅 related to ̅̅̅̅
𝐶𝐴?
̅̅̅̅ and 𝑅𝐴
̅̅̅̅. What can you
2. Measure the lengths of 𝐶𝑅
say about their lengths?
3. Other than being an altitude, how do you call ̅̅̅̅
𝐻𝑅
with respect to ̅̅̅̅
𝐶𝐴?
ACTIVITY 3 “CHOOSE THE TWO’S”
Which from among these pairs of congruent triangles
may be used to be able to construct the perpendicular
bisector of a line segment? Show how your choices
can be used.
Answer the following questions:
1. What is the common characteristic of the pairs of
congruent triangles that can be used to construct a
perpendicular bisector?
2. What kind of triangles can be used in constructing
perpendicular bisector?
3. Can you use two congruent scalene right triangles
to construct a perpendicular bisector? How?
In ̅̅̅̅
𝑃𝑇,
1. Draw two circles whose centers are at 𝑃 and 𝑇 using
the same opening of the compass.
2. Name the points of intersection of the two circles as
𝐴 and 𝑆. Connect points 𝐴 and 𝑆 using a line.
3. Connect points A and S to points P and T to form
∆𝑃𝑇𝐴 and ∆𝑃𝑇𝑆.
What can you say about ∆𝑃𝑇𝐴 and ∆𝑃𝑇𝑆?
What kind of triangle according to sides are ∆𝑃𝑇𝐴 and
∆𝑃𝑇𝑆?
̅̅̅̅ to 𝑃𝑇
̅̅̅̅?
What is the relation of 𝐴𝑆
153
5.Summarizing the
Lesson
What is a perpendicular bisector?
What kind of triangles are formed when constructing a
perpendicular bisector?
C. Assessment
∆𝑃𝑄𝑅 is an isosceles triangle. Construct the
̅̅̅̅ using its congruent
perpendicular bisector of.𝑃𝑅
triangle.
D. Agreement/
Assignment
Values Integration
1. Draw an isosceles obtuse triangle and name it ∆𝐽𝑂𝐸.
2. Cut out a triangle congruent to ∆𝐽𝑂𝐸.
3. Construct the three altitudes using its congruent
triangle.
4. Name the altitude which is also considered a
perpendicular bisector.
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require
additional
activities
for
remediation.
C. Did the remedial
lessons work? No.
of learners who
have caught up with
the lesson.
D. No. of learners who
continue to require
remediation.
E. Which
of
my
teaching strategies
worked well? Why
did it work?
F. What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
G. What innovation or
localized material/s
did I use/discover
which I wish to
share with other
teachers?
154
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisectors
School
Grade Level
GRADE 8
Teacher
Learning
Area
Quarter
MATHEMATICS
Time & Date
I.
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
THIRD
Week 10, Day 3
The learner demonstrates understanding of key concepts
of axiomatic structure of geometry and triangle
congruence.
The learner is able to communicate mathematical thinking
with coherence and clarity in formulating, investigating,
analyzing, and solving real-life problems involving
congruent triangles using appropriate and accurate
representations.
Applies triangle congruence to construct perpendicular
lines and angle bisectors.
Applying triangle
Bisectors)
III.
congruence
(Constructing
Angle
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning
Resources
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary
ACTIVITY 1
Activities/ Motivation
Given ∠𝐴𝐵𝐶, construct its angle bisector.
B. Presentation of the
Lesson
1. Problem Opener
ACTIVITY 2
155
2. Group Activity
3. Processing the
Answer
4. Reinforcing the
Skills
5.Summarizing
the Lesson
C. Assessment
D. Agreement/
Assignment
V.
REMARKS
VI.
REFLECTION
VII.
OTHERS
A. No. of learners who
earned 80% on the
formative
assessment
B. No. of learners who
require additional
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅ . Perform
In the figure, 𝐴𝐵
or answer the following:
1. Connect points A and D
and points D and C.
2. Name the two triangles
formed.
3. What can you say about
the triangles? (Let students prove their
congruence or show their congruence using
protractor or ruler.)
ACTIVITY 3
Construct the angle bisector of the given angle.
TASKS:
1. Name the vertex as E. Locate a point on each
side of the obtuse angle such that, they are
equidistant from point E. Name them as T and A.
2. Locate 5 points on the angle bisector and name
them P, Q, R, S and T.
3. Connect all points on the angle bisector to points
T and A.
Answer the following based from Activity 3:
1. Which of the segments formed are congruent?
2. Which of the triangles formed are congruent?
3. Describe how the congruent triangles are
positioned when constructing the angle bisector.
1. Draw an angle named ∠𝑇𝐸𝐴.
2. Construct its angle bisector.
3. Show a pair of congruent triangles formed when
constructing the angle bisector.
How can congruent triangles be used in constructing an
angle bisector?
Given ∠𝐵𝐻𝐴, construct the angle bisector and show the
congruent triangles formed. Name these triangles.
Construct the angle bisector of each angle of ∆𝑀𝑋𝑇.
156
C.
D.
E.
F.
G.
activities
for
remediation.
Did the remedial
lessons work? No.
of learners who
have caught up
with the lesson.
No. of learners who
continue to require
remediation.
Which
of
my
teaching strategies
worked well? Why
did it work?
What difficulties did
I encounter which
my principal or
supervisor can help
me solve?
What innovation or
localized material/s
did I use/discover
which I wish to
share with other
teachers?
157
(M8GE-IIIi-j-1) - Applies Triangle Congruence to Construct
Perpendicular Lines and Angle Bisectors
School
Grade Level
GRADE 8
Teacher
Learning
Area
Quarter
MATHEMATICS
Time
Date
I.
&
OBJECTIVES
A. Content
Standard
B. Performance
Standard
C. Learning
Competencies/
Objectives
II.
CONTENT
THIRD
Week 10, Day 4
The learner demonstrates understanding of key
concepts of axiomatic structure of geometry and
triangle congruence.
The learner is able to communicate mathematical
thinking with coherence and clarity in formulating,
investigating, analyzing, and solving real-life problems
involving congruent triangles using appropriate and
accurate representations.
Applies triangle congruence to construct perpendicular
lines and angle bisectors.
Applying
triangle
congruence
(Constructing
Perpendicular Lines and Angle Bisectors)
III.
LEARNING
RESOURCES
A. References
1. Teacher’s
Guide pages
2. Learner’s
Materials
pages
3. Textbook
pages
4. Additional
Materials from
Learning
Resource (LR)
portal
B. Other Learning
Resources
IV.
PROCEDURE
ADVANCED LEARNERS
AVERAGE LEARNERS
A. Preliminary
- Recall on Constructing Perpendicular Lines and Angle
Activities/ Motivation
Bisectors
- Recall on how Congruent Triangles are formed when
constructing Perpendicular Lines and Angle Bisector
B. Presentation of
the Lesson
1. Problem
Opener
WORKSHEET (Constructing Perpendicular Lines and
Angle Bisector)
158
Name: __________________________________ Gr. And Section: __________________
WORKSHEET
Constructing Perpendicular Lines and Angle Bisectors
Materials: Compass and Straight Edge
A. Construct the perpendicular lines of the given line segments passing through the
given point and show the congruent triangles formed.
1.
2.
B. Construct the angle bisector the given angle and show the congruent triangles
formed.
159
Third Quarter Pre-Test in Mathematics 8
Direction: Understand each question/problem properly, then select the best answer
from the given choices by writing only the letter corresponding to it.
1. How many geometric ideas are referred to as “undefined terms”?
a. 0
b. 1
c. 2
d. 3
2. Which of these is the best illustration of a point?
a. a ball
b. the moon
c. tip of a pen
d. corner of a table
3. Avelino walks home from school. Tracing his way one day, he noticed that after
walking for a seemingly straight path he then turned slightly to the right to reach the
house. Which of these is best illustrated?
a. ray
b. line
c. angle
d. triangle
4. The part of a line with an endpoint and extends indefinitely in one direction is a ___.
a. ray
b. line
c. half-line
d. line segment
5. It is a mathematical statement wherein the truthfulness is still to be established.
a. axiom
b. theorem
c. corollary
d. postulate
6. Two points determine a line. This statement is considered a ______.
a. axiom
b. guess
c. theorem
d. postulate
7. Two triangles are _____ if their vertices can be paired so that their corresponding
angles and corresponding sides are congruent.
a. one
b. equal
c. similar
d. congruent
8. What is the sum of the measures of the angles of a triangle?
a. 900
b. 1800
c. 2700
d. 3600
9. Based on similar markings, what can be said about the two triangles?
a. The two triangles are similar.
b. The two triangles are different.
c. The two triangles are congruent.
d. The two triangles overlapped each other.
10. It is a mathematical statement which is already accepted to be true.
a. axiom
b. theorem
c. corollary
d. postulate
11. Triangles are classified into how many classifications?
a. 1
b. 2
c. 3
d. 7
12. How many triangle congruence postulates were there to prove triangle
congruence?
a. 1
b. 2
c. 3
d. 4
13. Which of these statements can be proven by triangle congruence?
a. A diagonal of a rhombus divides it into congruent triangles.
b. Every angle bisector of a quadrilateral divides it into congruent triangles.
c. Every altitude of an isosceles triangle divides it into congruent triangles.
d. Congruent triangles can be formed from any right triangles when cut
along the right angle.
14. Which figure illustrates ASA Congruence Postulate?
a.
b.
c.
d.
160
𝑇𝑦𝑝𝑒
𝑇𝑦𝑝𝑒
𝑇𝑦𝑝𝑒 𝑇𝑦𝑝𝑒
𝑇𝑦𝑝𝑒 𝑇𝑦𝑝𝑒
15. How many corresponding parts of two triangles must be proven to be congruent
before the two triangles can be congruent?
a. 4
b. 3
c. 2
d. 1
16. Which of the following best describe the creation of a plane?
a. mat making
b. road widening
c. cloud seeding
d. wall painting
17. Which other corresponding parts must be congruent in order that the two triangles
are congruent by SAS Congruence Postulate?
L
S
a. ̅̅̅̅
𝐿𝐴 ≅ ̅̅̅̅
𝑆𝑁
b. ̅̅̅̅̅
𝐿𝑊 ≅ ̅̅̅̅
𝑆𝑁
̅
̅
c. ̅̅̅̅̅
𝐴𝑊 ≅ 𝑆𝐼
d. ̅̅̅̅
𝐴𝐿 ≅ 𝐼𝑆
A
W I
N
18. These materials are used to make geometric constructions.
a. ruler and pencil
b. chalk and board
c. triangle and protractor
d. compass and straightedge
19. Which triangle congruence postulate uses two angles and a non-included side?
a. SSS
b. SAA
c. SAS
d. ASA
20. An angle bisector of any angle of a triangle is always perpendicular to the opposite
side. This statement is ______.
a. true
b. false
c. baseless
d. sometimes true
21. Avelino and Pedro are each making a triangle. They want that the triangles are
congruent. Now, Pedro took two pieces of sticks each measuring 25 cm and 50
cm, respectively. Then, formed an angle with it measuring 35 degrees before
connecting the third side. Should Avelino do the same too?
a. no
b. yes
c. maybe
d. he can try
22. For two triangles to be congruent, how many corresponding sides must be
congruent?
a. 1
b. 2
c. 3
d. none
23. What angle is formed by perpendicular lines?
a. acute angle
b. obtuse angle
c. vertex angle
d. right angle
24. If the hypotenuses of two congruent right triangles are exactly attached, then the
hypotenuse become _______ of the angle formed by the adjacent acute angles of
the two right triangles.
a. a side
b. a divisor
c. a common side
d. an angle bisector
25. Given the isosceles trapezoid at the right, which corresponding parts of the triangle
ADC and triangle BCD can be proven easily as congruent?
A
B
a. three corresponding sides
b. any two corresponding sides and its included angle
c. any two corresponding angles and its included side
d. any two corresponding angles and a corresponding side C
D
26. A triangle which is not divided into congruent triangles by any of its median is not
an equilateral triangle. This statement is considered _____.
a. false
b. true
c. a theorem
d. a corollary
27. What can you say about this statement?
“It is possible for a triangle to be both acute and scalene. “
a. It is untrue.
b. It is true.
c. It will never happen.
d. It is hard to illustrate.
28. What are the kinds of angle?
a. acute, chronic, malignant
b. acute, obvious, correct
c. acute, obtuse, right
d. simple, average, difficult
161
29. To every angle there correspond a unique number between 0 0 and 1800 called
_____ of the angle.
a. the size
b. the opening
c. the measure
d. the associated number
30. Which pair of angles are always congruent?
a. linear pair
b. vertical angles
c. complementary angles
d. supplementary angles
31. Two ______ lines are ______ if and only if they _______.
a. coplanar, parallel, do not intersect
b. parallel, coplanar, do not intersect
c. coplanar, do not intersect, parallel
d. parallel, do not intersect, coplanar
32. Which of the following supports the existence of a mathematical system?
i. Mathematics cannot exist with numbers only.
ii. Part of the mathematical system are the properties and deduction of
geometry, measurement, and statistics.
iii. Illustrations, equations, and solutions are needed to establish
truthfulness.
a. i and iii
b. i and ii
c. ii and iii
d. i, ii, and iii
33. It illustrates two half-planes which are coplanar and do not have a common edge.
a.
b.
c.
d.
34. Based on similar markings which are the corresponding congruent parts of the two
triangles?
a. ∠ A ≅ ∠ K
∠C ≅ ∠B
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐵𝐶
b. ∠ B ≅ ∠ C
∠C ≅ ∠B
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐵𝐶
c. ∠ ACB ≅ ∠ KBC
∠ BCK ≅ ∠ CBA
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐵𝐶
d. ∠ ABC ≅ ∠ BCK
∠ KBC ≅ ∠ BAC
̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐵𝐾
A
B
C
35. Give the congruence statement that can prove the congruence of the two triangles
below.
a. SSS
b. SAS
c. ASA
d. SAA
36. Two sides and its non-included angle of one triangle are equal in measure with two
sides and its non-included angle of another triangle. Which of these statements
will be true?
a. The two triangles are congruent.
b. The triangles have same size and shape.
c. The two triangles will not be congruent.
d. The two triangles may or may not be congruent.
162
K
For 37-38: Refer to the following informations:
Given: ̅̅̅̅
𝑃𝐿 ≅ ̅̅̅̅
𝐴𝐿
∠ PLN and ∠ ALN are right angles
Prove: ∆ PLN ≅ ∆ ALN
37. Which of the figures below will suit the information above?
a.
P
b.
P
c. P
L
L
N
N
A
L
N
A
A
38. ∆ PLN and ∆ ALN can be proven congruent by which postulate?
a. LA
b. HyA
c. HyL
For 39 – 43. Refer to the figure below.
A
C T
I
G
d.
P
A
N
L
d. LL
̅̅̅̅ ≅ 𝐺𝑁
̅̅̅̅
Given: 𝐴𝑁
̅̅̅
̅̅̅
𝐴𝐼 ≅ 𝐼𝐺
N
39. If AN = 40 cm., how long is GN?
a. 20 cm
b. 30 cm
c. 40 cm
d. 50 cm
40. Which triangle is congruent to triangle TIA?
a. Δ TIG
b. Δ ATI
c. Δ TGI
d. Δ TNG
41. Which segment bisects angle ANG?
̅̅̅̅
̅̅̅̅
̅̅̅̅
̅̅̅
a. 𝐴𝑇
b. 𝑁𝐼
c. 𝐶𝑇
d. 𝑇𝐼
42. If segment AC and segment CG are drawn, then ____ will bisect angle ACG and
segment AG.
̅̅̅.
̅̅̅̅
̅̅̅̅
̅̅̅̅
a 𝑇𝐼
b. 𝑁𝐼
c. 𝐴𝑇
d. 𝑇𝐶
43. The group of points below are collinear except _____.
a. A, T, G
b. T, I, N
c. N, C, T
d. A, I, G
44. What idea will complete the meaning of the sentence below?
“B is between A and C if and only if A, B, and C lie on one line, and _______.”
a. /AB/ + /BC/ = /AC/
c. /AC/ + /BC/ = /AB/
b. /AB/ + /AC/ = /BC/
d. /BC/ + /BC/ = /AC/
45. Which angles in the illustrations are adjacent?
a.
b.
c.
d.
46. The measure of angle A is 5m + 7 while its complement angle J measures 3m – 5.
What is the value of “m” ?
a. 44
b. 45
c. 46
d. 48
47. How is “distance from a point to a line” defined?
a. It is the length of any two points on the line.
b. The distance from a point to the line is constant.
c. It is the measure of the space between the point and the line.
d. The length of the perpendicular segment from the point to the line.
48. These angles are always congruent. What are these angles?
a. acute angles
b. right angles
c. obtuse angles
d. vertex angles
163
For 49-50. Refer to the figure below.
7 + 4k
S
9 +2k
49. What is the value of k ?
a. 4
b. 3
50. What is the perimeter of triangle PSM?
a. 32
b. 30
______
*** rbjr.2019
164
P
5
A L
5
M
c. 2
d. 1
c. 26
d. 18
Third Quarter Pre-Test in Mathematics 8
(Answer Key)
1
d
14
c
27
b
40
a
2
c
15
b
28
c
41
b
3
c
16
a
29
c
42
d
4
a
17
d
30
b
43
d
5
b
18
d
31
a
44
a
6
d
19
b
32
d
45
b
7
d
20
d
33
a
46
a
8
b
21
b
34
c
47
d
9
c
22
c
35
d
48
b
10
d
23
c
36
c
49
d
11
b
24
d
37
c
50
a
12
d
25
a
38
d
13
a
26
b
39
c
165
Third Quarter Post Test in Mathematics 8
Direction: Understand each question/problem properly, then select the best answer
from the given choices by writing only the letter corresponding to it.
1. How many geometric ideas are referred to as “undefined terms”?
a. 3
b. 2
c. 1
d. 0
2. Which of these is the best illustration of a point?
a. a ball
b. the moon
c. wall clock
d. grain of sand
3. Pedro walks home from work. One day, he noticed that after walking in a seemingly
straight path he then turned slightly to the left to reach the house. Which of these is
best illustrated?
a. curve
b. angle
c. line
d. triangle
4. The part of a line with two endpoints is a ______.
a. ray
b. line
c. half-line
d. line segment
5. Mathematical statement that need to be proven.
a. axiom
b. corollary
c. theorem
d. postulate
6. If two points belong to a plane, then the line determined by the two points is
contained in the plane. This statement is considered a ______.
a. postulate
b. theorem
c. guess
d. axiom
7. Two triangles are congruent if their ______can be paired so that their corresponding
angles and corresponding sides are congruent.
a. sides
b. angles
c. vertices
d. shapes
8. The sum of the measures of the angles of an scalene triangle is 1800. How about the
sum of the measures of the angles of an obtuse triangle?
a. 900
b. 1800
c. 2700
d. 3600
9. Based on similar markings, which is not true about the two triangles?
a. The two triangles have the same size and shape.
b. The two triangles can overlapped each other.
c. The two triangles will not fit each other.
d. The two triangles are congruent.
10. It is a mathematical statement which does not need to be proven.
a. axiom
b. theorem
c. corollary
d. postulate
11. How many triangles are there according to angles?
a. 1
b. 2
c. 3
d. 4
12. How many triangle congruence postulates were there to prove triangle congruence?
a. 4
b. 3
c. 2
d. 1
13. Which of these statements can be proven by triangle congruence?
a. A diagonal of a rhombus divides it into congruent triangles.
b. Every angle bisector of a quadrilateral divides it into congruent triangles.
c. Every altitude of an isosceles triangle divides it into congruent triangles.
d. Congruent triangles can be formed from any right triangles when cut along
the right angle.
14. Which figure illustrates ASA Congruence Postulate?
a.
b.
d.
𝑇𝑦𝑝𝑒
c.
𝑇𝑦𝑝𝑒
𝑇𝑦𝑝𝑒 𝑇𝑦𝑝𝑒
𝑇𝑦𝑝𝑒 𝑇𝑦𝑝𝑒
166
15. Given the isosceles trapezoid at the right, which corresponding parts of the triangle ADC
and triangle BCD can be proven easily as congruent?
a. three corresponding sides
A
B
b. any two corresponding sides and its included angle
c. any two corresponding angles and its included side
d. any two corresponding angles and a corresponding side
C
D
16. What are the kinds of angle?
a. acute, chronic, malignant
b. acute, obvious, correct
c. simple, average, difficult
d. acute, obtuse, right
17. Which other corresponding parts must be congruent in order that the two triangles
are congruent by ASA Congruence Postulate?
L
S
̅̅̅̅ ≅ ̅̅̅̅
a. 𝐿𝐴
𝑆𝑁
b. ∠𝑊 ≅ ∠𝑁
̅
c. ∠𝐿 ≅ ∠𝑆
d. ̅̅̅̅
𝐴𝐿 ≅ 𝐼𝑆
A
W I
N
18. These materials are used to make geometric constructions.
a. ruler and pencil
b. chalk and board
c. triangle and protractor
d. compass and straightedge
19. Which triangle congruence postulate uses two angles and an included side?
a. SSS
b. SAA
c. SAS
d. ASA
20. An angle bisector of the right angle of a right triangle is always perpendicular to the
opposite side. This statement is ______.
a. true
b. false
c. baseless
d. sometimes true
21. Ton and Mon are each making a triangle. They want that the triangles are
congruent. Now, Ton took two pieces of sticks each measuring 25 cm and 50 cm,
respectively. Then, formed an angle with it measuring 50 degrees before
connecting the third side. Should Mon do the same too?
a. no
b. yes
c. maybe
d. he can try
22. For two triangles to be congruent, how many corresponding sides must be
congruent?
a. 3
b. 2
c. 1
d. none
23. If two lines formed a right angle, then the two lines are _____?
a. the same
b. not coplanar
c. perpendicular
d. parallel
24. If the hypotenuses of two congruent right triangles are exactly attached, then the
hypotenuse become _______ of the angle formed by the adjacent acute angles of
the two right triangles.
a. a side
b. a divisor
c. a common side
d. an angle
bisector
25. How many corresponding parts of two triangles must be proven to be congruent
before the two triangles can be congruent?
a. 1
b. 2
c. 3
d. 4
26. A triangle which is not divided into congruent triangles by any of its median is not
an equilateral triangle. This statement is considered _____.
a. false
b. true
c. a theorem
d. a corollary
27. What can you say about this statement?
“It is possible for a triangle to be both obtuse and scalene. “
a. It is untrue.
b. It is true.
c. It will never happen.
d. It is hard to illustrate.
28. Which of the following does not describe the creation of a plane?
a. tiling
b. road making
c. cloud seeding
d. wall painting
29. If x is the measure of an obtuse angle then ______.
a. 00 < x < 1800 b. 00 > x > 1800
c. 00 < x > 1800 d. 00 > x < 1800
167
30. Give the congruence statement that can prove the congruence of the two triangles
below.
a. SSS
b. SAS
c. ASA
d. SAA
31. Two ______ lines are ______ if and only if they _______.
a. coplanar, parallel, do not intersect
b. parallel, coplanar, do not intersect
c. coplanar, do not intersect, parallel
d. parallel, do not intersect, coplanar
32. Which of the following supports the existence of a mathematical system?
i. Mathematics cannot exist with numbers only.
ii. Part of the mathematical system are the properties and deduction of
geometry, measurement, and statistics.
iii. Illustrations, equations, and solutions are needed to establish truthfulness.
a. i and iii
b. i and ii
c. ii and iii
d. i, ii, and iii
33. It illustrates two half-planes which are parallel.
a.
b.
c.
d.
34. Based on similar markings which are the corresponding congruent parts of the two
triangles?
a. ∠ A ≅ ∠ K
b. ∠ B ≅ ∠ C
A
∠C ≅ ∠B
∠C ≅ ∠B
̅̅̅̅
̅̅̅̅
𝐵𝐶 ≅ ̅̅̅̅
𝐵𝐶
𝐵𝐶 ≅ ̅̅̅̅
𝐵𝐶
B
35.
36.
37.
38.
c. ∠ ACB ≅ ∠ KBC
d. ∠ ABC ≅ ∠ BCK
C
∠ BCK ≅ ∠ CBA
∠ KBC ≅ ∠ ACB
̅̅̅̅ ≅ 𝐵𝐶
̅̅̅̅
̅̅̅̅ ≅ 𝐵𝐾
̅̅̅̅
𝐵𝐶
𝐴𝐶
Which pair of angles have a total measure equal to 1800?
a. linear pair
b. vertical angles
c. complementary angles
d. supplementary angles
The corresponding angles of two triangles are congruent. Which of these
statements is true?
a. The two triangles are congruent.
b. The triangles have same size and shape.
c. The two triangles will not be congruent.
d. The two triangles may or may not be congruent.
What idea will complete the meaning of the sentence below?
“B is between A and C if and only if A, B, and C lie on one line, and _______.”
a. /AB/ - /BC/ = /AC/
b. /AB/ + /AC/ = /BC/
c. /AB/ + /BC/ = /AC/
d. /AC/ - /BC/ = /AC/
Which angles in the illustrations are adjacent?
a.
b.
c.
d.
39. The measure of angle A is 5m – 42 while its supplement angle J measures 3m – 34.
What is the value of “m” ?
a. 44
b. 37
c. 32
d. 28
40. What are the angles that are always congruent?
a. acute angles
b. right angles
c. obtuse angles
d. vertex angles
168
K
41. How is “distance from a point to a line” defined?
a. The length of the perpendicular segment from the point to the line.
b. It is the measure of the space between the point and the line.
c. The distance from a point to the line is constant.
d. It is the length of any two points on the line.
For 42 – 46. Refer to the figure below.
A
C
T
I
G
N
Given: ̅̅̅̅
𝐴𝑁 ≅
̅̅̅
𝐴𝐼 ≅
̅̅̅̅
𝐺𝑁
̅̅̅
𝐼𝐺
42. If AN = 37 cm., how long is GN? a. 20 cm b. 30 cm
c. 37 cm
d. 45 cm
43. Which triangle is congruent to triangle TNA?
a. Δ TIG
b. Δ ATI
c. Δ TGI
d. Δ TNG
̅̅̅̅?
44. Which segment is not perpendicular 𝐴𝑇
̅̅̅
a. ̅̅̅
𝑇𝐼
b. ̅𝑁𝐼
c. ̅̅̅̅
𝐶𝑇
d. ̅̅̅̅
𝑇𝑁
45. If segment AC and segment CG are drawn, then ____ will bisect angle ACG and
segment
AG.
̅̅̅
̅̅̅
a 𝑇𝐼.
b. ̅𝑁𝐼
c. ̅̅̅̅
𝐴𝑇
d. ̅̅̅̅
𝑇𝐶
46. The group of points below are collinear except _____.
a. A, T, G
b. T, I, N
c. N, G, T
d. N, C, T
For 47-48. Refer to the figure below.
A
47. What is the value of k ?
a. 4
48. What is the perimeter of triangle PAM?
a. 42
b. 58
P
5
L
5
M
b. – 3
c. 2
d. – 1
c. 69
d. 96
For 49-50: Refer to the following informations.
Given: F is on the side of ∆ LIE ; ̅̅̅
𝐼𝐹 ≅ ̅̅̅̅
𝐹𝐸
∠ LFI and ∠ LFE are right angles
Prove: ∆ LIF ≅ ∆ LEF
49. Which of the figures below will suit the information above?
a.
F
b.
L
c.
I
E
F
I
I
L
d.
L
F
E
E
F
I
E
50. ∆ LIF and ∆ LEF can be proven congruent by which postulate?
a. LA
b. LL
c. HyL
____
*** rbjr.2019
169
d. HyA
L
Third Quarter Post-Test in Mathematics 8
(Answer Key)
1
a
14
c
27
b
40
b
2
d
15
a
28
c
41
a
3
b
16
d
29
a
42
c
4
d
17
b
30
b
43
d
5
c
18
d
31
a
44
b
6
a
19
d
32
d
45
d
7
c
20
d
33
b
46
c
8
b
21
b
34
c
47
a
9
c
22
a
35
d
48
d
10
d
23
c
36
d
49
c
11
d
24
d
37
c
50
b
12
a
25
c
38
b
13
a
26
b
39
c
170
Mathematics 8
Third Quarterly Examination
Table of Specifications
COGNITIVE PROCESS DIMENSIONS
Analyzing
Evaluating
Creating
171
No.
of
Items
Applying
Total
%
Understanding
1. Describes mathematical system
2. Illustrates the need for an axiomatic structure of a mathematical system in
general, and in Geometry in particular: (a) defines terms; (b) undefined terms; (c)
postulates; and (d) theorems
3. Illustrates triangle congruence
4. Illustrates the SAS, ASA, SSS, and SAA
5. Identifies and solves corresponding parts of congruent triangles
6. Proves congruence of triangles
7. Proves statements using triangle congruence
8. Applies triangle congruence to identify, show, or construct perpendicular lines
and angle bisectors
No. of
Days
Remembering
Competencies
2
5
3
1
1
1
10
25
13
4
3
3
1
1
1
2
6
4
4
4
5
15
10
10
10
2
7
5
5
5
1
2
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
8
20
10
3
2
2
1
1
1
40
100
50
15
10
10
5
5
5
1
1