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Department of Education-Region III
TARLAC CITY SCHOOLS DIVISION
Juan Luna St., Sto. Cristo, Tarlac City 2300
Email address: tarlac.city@deped.gov.ph/ Tel. No. (045) 470 - 8180
MATHEMATICS
Quarter 4: Week 3 (Week 4)
Learning Activity Sheet
Mathematics 8
Name: _________________________________
Section: ______________________
Date: ___________
Quarter 4 - Week 3 (Week 4)
Properties of Parallel Lines cut by a Transversal
Background Information
Line is a one-dimensional figure that consists of points. It has no thickness and that extends
in different directions.
Line AB can be written as AB
B
A
Other examples of lines are:
a. Road
b. Chopstick
c. Baseball bat
Key symbols:
1.
2.
3.
4.
∠
≅
∥
- symbol for line
- symbol for angle
- congruent symbol and read as “is congruent to”
- symbol for parallel lines
There are five (5) common lines in Proving the Properties of Parallel Lines cut by a
Transversal: (1) coplanar lines; (2) parallel lines; (3) intersecting lines; (4) perpendicular lines; (5)
transversal lines.
1. Coplanar lines are lines lying on the
1.
Coplanar
lines are lines lying on the same plane.
same
plane.
2. Parallel lines are coplanar lines that do
not intersect.
r
k
s
j
s
s
3. Intersecting lines are coplanar lines that
have a point in common.
4. Perpendicular lines are lines that
intersect at right angles
m
m
s
n
o
s
s
p
s
5. Transversal line is a line intersecting two or more coplanar lines that are usually parallel.
a - transversal line
b
c
s
1
A Transversal creates different angle pairs. To illustrate, let f be a transversal line of lines g
and h.
f
c
s
1
3
2s
g
4
5 6
7 8
h
Pair of Corresponding angles are:
∠1 and ∠5, ∠2 and ∠6
∠3 and ∠7, ∠4 and ∠8
Interior angles are:
∠3, ∠4, ∠5, and ∠6
Exterior angles are:
∠1, ∠2, ∠7, and ∠8
Pairs of alternate interior angles:
∠3 and ∠6
∠4 and ∠5
Pairs of alternate exterior angles:
∠1 and ∠8
∠2 and ∠7
Pairs of same-side interior angles:
∠3 and ∠5
∠4 and ∠6
Pairs of same-side exterior angles:
∠1 and ∠7
∠2 and ∠8
Based on the given example, the relationship between angle pairs formed by parallel lines
cut by a transversal line are stated in the following postulates and theorems.
Corresponding Angles Postulate (CAP): If two parallel lines are cut by a transversal, then
each pair of corresponding angles is congruent.
a
If ∠1 ≅ ∠5, therefore b ∥ c.
b
1 2
3 4
c
Take note: Congruent means similar, the same
or identical.
6
8
7
5
THEOREM 8.1 Alternate Interior Angles Theorem: If two parallel lines are cut by a
transversal, then alternate interior angles are congruent.
a
1 2
3 4
5
Given: b ∥ c
b
c
Prove: ∠4 ≅ ∠5
6
7 8
2
Statement
1.
2.
3.
4.
Reasons
b∥c
∠4 ≅ ∠1
∠1 ≅ ∠5
∠4 ≅ ∠5
1.
2.
3.
4.
Given
Vertical Angles Theorem
Corresponding Angles Postulate
Transitive Property of Angle
Congruence
THEOREM 8.2 Alternate Exterior Angles Theorem: If two parallel lines are cut by a
transversal, then alternate exterior angles are congruent.
a
b
1 2
3 4
5
7
Given: b ∥ c cut by transversal a
c
Prove: ∠1 ≅ ∠8
6
8
Statement
1.
2.
3.
4.
Reasons
b∥c
∠1 ≅ ∠5
∠5 ≅ ∠8
∠1 ≅ ∠8
1.
2.
3.
4.
Given
Corresponding Angles Postulate
Vertical Angles Theorem
Transitive Property of Angle
Congruence
THEOREM 8.3 Same-Side Interior Angles Theorem: If two parallel lines are cut by a
transversal, then the interior angles on the same side of the transversal are supplementary.
a
1 2
3 4
5
7
Given: b ∥ c cut by transversal a
b
c
Prove: ∠3 and ∠5 are supplementary
Take note: Two angles are supplementary when
they add up to 180°.
6
8
Statement
1.
2.
3.
4.
Reasons
b∥c
∠1 and ∠3 form a linear pair.
∠1 and ∠3 are supplementary.
m∠1 + m∠3 = 180°
1.
2.
3.
4.
3
Given
Definition of linear pair
Linear Pair Postulate
Definition of supplementary angles
5. ∠1 ≅ ∠5
6. m∠1 = m∠5
7. m∠3 + m∠5 = 180°
5. Corresponding Angles Postulate
6. Definition of Congruent Angles
7. Statements 4 and 6, and Substitution
Property of Equality
8. Definition of supplementary angles.
8. ∠3 and ∠5 are supplementary.
Based on the postulate and theorems, let us solve the following examples. Let m and n are
cut by a transversal k. Find the value of x that makes m ∥ n.
Examples
a. ∠3 and ∠6, are alternate interior angles, prove that m∠1 = 6𝑥 + 4 is congruent to the
m∠5 = 2𝑥 + 12.
Solution:
Checking:
6𝑥 + 4 = 2𝑥 + 12
−2𝑥 + 6𝑥 + 4 = 2𝑥 + 12 − 2𝑥
4𝑥 + 4 = 12
−4 + 4𝑥 + 4 = 12 − 4
4𝑥 8
=
4
4
𝑥=2
Given
6(2) + 4 = 2(2) + 12
12 + 4 = 4 + 12
16 = 16
Subtraction property of equality
Simplify
Subtraction property of equality
Division property of equality
Therefore, the m∠1 = 6𝑥 + 4 is
congruent to the m∠5 = 2𝑥 + 12.
b. ∠3 and ∠5, are same-side interior angles, prove m∠1 = −8𝑥 + 5 is congruent to the
m∠5 = 13𝑥 + 20.
Solution:
Checking:
8𝑥 − 5 = 13𝑥 − 20
−13𝑥 + 8𝑥 − 5 = 13𝑥 − 20 − 13𝑥
−5𝑥 − 5 = −20
5 + −5𝑥 − 5 = −20 + 5
−5𝑥 −15
=
−5
−5
𝑥=3
Given
8(3) − 5 = 13(3) − 20
24 − 5 = 39 − 20
19 = 19
Subtraction property of equality
Simplify
Subtraction property of equality
Division property of equality
Therefore, the m∠1 = −8𝑥 + 5 is
congruent to the m∠5 = 13𝑥 + 20.
c. ∠3 and ∠6, are alternate exterior angles, prove m∠3 = 150° is congruent to the m∠6
= 3𝑥 − 15.
Solution:
3𝑥 − 15
15 + 3𝑥 − 15
3𝑥
3
𝑥
Checking:
= 150
= 150 + 15
165
=
3
= 55
3(55) − 15 = 150
165 − 15 = 150
150 = 150
Given
Addition property of equality
Division property of equality
Therefore, the m∠3 = 150° is
congruent to the m∠6 = 3𝑥 − 15.
Learning Competency
Proves properties of parallel lines cut by a transversal. (8GE-IVd-1)
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General Directions: Read carefully the directions indicated in the following activity, then, provide
what is needed or asked in each item. Write the answers on a separate sheet of paper.
Activity 1: For items 1-5, refer to the figure below. Tell whether each statement is TRUE or FALSE.
Write TRUE if the statement is correct, otherwise change the underlined word to make the
q
statement true.
1
____________1. ∠1 and ∠5 are corresponding angles.
____________2. ∠4 and ∠6 are pairs of alternate exterior angles.
____________3. ∠3 and ∠5 are pairs of alternate interior angles.
____________4. ∠2 and ∠8 are same-side exterior angles.
____________5. ∠1, ∠2, ∠7, ∠8 are interior angles.
4
2
3
r
5 6
8 7
Activity 2: Do as indicated.
A. State each postulate and theorem.
1. Corresponding Angles Postulate (CAP) – _____________________________________
2. Theorem 8.2 – __________________________________________________________
3. Theorem 8.1 – __________________________________________________________
4. Theorem 8.3 – __________________________________________________________
B. Let p and q are cut by a transversal n. Find the value of x that makes p ∥ q. Solve problem.
∠4 and ∠6, are alternate interior angles, prove that m∠4 = 𝟕𝒙 + 𝟓𝟐 is congruent to the
m∠6 = 𝟏𝟒𝒙 + 𝟏𝟎. Show your solution.
Activity 3: For items 1-5, refer to the figure from activity 1.
Given: r ∥ s, m∠7 = 102°.
1. Find m∠3 = _________
2. Find m∠4 = _________
3. Find m∠6 = _________
4. Find m∠2 = _________
5. Find m∠5 = _________
Activity 4: Complete the proof of the Alternate Interior Angles Theorem.
l
Given: m ∥ n
Prove: ∠3 ≅ ∠6
m
1 2
3 4
5
n
6
7 8
Statement
1.
2.
3.
4.
Reasons
_______
∠3 ≅ ∠2
__________
__________
1.
2.
3.
4.
5
__________________________
__________________________
Corresponding Angles Postulate
Transitive Property of Angle
Congruence
s
Activity 5: Use a clean bond paper. Draw the floor plan of your house. From the illustration you
made, highlight (using red marker or crayon) the parallel lines cut by a transversal. Be creative and
artistic!
Project Rubric:
Criteria
Mathematical
Relevance
Neatness and Visual
Appeal
Outstanding
(10-9 pts.)
Shows very good
understanding of
parallel lines cut by
transversal.
Good
(8-7 pts.)
Shows a good
understanding of
parallel lines cut by
transversal.
Design of the floor
plan is very neat with
clear illustration of the
house.
Design of the floor
plan is neat with
almost clear
illustration of the
house.
Novice
(6-5 pts.)
The floor plan is
incomplete and does
not show
understanding of
parallel lines cut by
transversal.
Design of the floor
plan is incomplete and
does not show a clear
illustration of the
house.
Reflection
Give 3 examples of real-life situations where parallel lines are cut by transversal.
__________________________________________________________________________________
________________________________________________________________________________________
________________________________________________________________________________________
References:
Oronce, Orlando A., and Mendoza , Marilyn O.., Exploring Math Textbook: Department of
Education- Bureau of Learning Resources(DepEd- BLR), Pasig City, Philippines 1600, 360-381.
Nivera, Gladys C., Grade 8 Mathematics Pattern and Practicalities: Don Bosco Press Inc.,
Antonio Arnaiz cor. Chino Roces Avenues, Makati City, Philippines, 2018, 384-392.
Arlene A. Pascasio, et.al, Math Ideas and Life Applications: Abiva Publishing House, Inc.,
Abiva Bldg., 851 G. Araneta Ave., 1113 Quezon City, Philippines, 2013, 394-404.
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Answer key:
Activity 1:
1.
2.
3.
4.
5.
TRUE
Alternate Interior Angles
TRUE
TRUE
Exterior Angles
Activity 2:
A.
1. Corresponding Angles Postulate (CAP): If two parallel lines are cut by a transversal, then
each pair of corresponding angles is congruent.
2. THEOREM 8.2 Alternate Exterior Angles Theorem: If two parallel lines are cut by a
transversal, then alternate exterior angles are congruent.
3. THEOREM 8.1 Alternate Interior Angles Theorem: If two parallel lines are cut by a
transversal, then alternate interior angles are congruent.
4. THEOREM 8.3 Same-Side Interior Angles Theorem: If two parallel lines are cut by a
transversal, then the interior angles on the same side of the transversal are supplementary.
B. 𝑥 = 6
Activity 3:
1.
2.
3.
4.
5.
102°
78°
78°
78°
102°
Activity 4: Answers may vary.
Activity 5: Answers may vary.
Prepared by:
Ian Joseph T. Supan
Substitute Teacher I
Flordeliza W. Pascua
Teacher I
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