OPENING ACTIVITY
GEOMETRY
JEOPARDY
A REVIEW ABOUT
COMPLEMENTARY
ANGLES &
SUPPLEMENTARY
ANGLES
Topic: Angle Pairs
What are complementary
angles?
What are supplementary
angles?
Consider the following:
Complementary
Angles
-Are two angles that
together make a
right angle.
The measures of
the two angles
must add up to
90°.
A
D
30º
60º
B
C
m  ABC = m  ABD + m  CBD
90 = 30 + 60
 ABD and  CBD are
COMPLEMENTARY
ANGLES
Consider the following:
R
Two 45º angle are
Complementary.
D
45º
 RPD and  QPD are
Complementary angles.
45º
P
Q
Can we say that  B and  Q are
Consider
this
figure
Complementary angles?
B
70º
Q
20º
 B and  Q are Complementary
angles.
 B is a complement to  Q.
 Q is a complement to  B.
Supplementary Angles
Are two angles that together form
one-half of a complete rotation—
that is, 180°.
 The measures of two
supplementary angles, therefore,
must add up to 180 when added
together.
 The supplementary angle of a 50°
angle, for example, is a 130°.

What can
you say the
about
the angle sum
Consider
following:
measure of  RPD and  QPD ?
D
145º
R
35º
P
m RPD + m  QPD = 180.
Therefore,  RPD and  QPD are
supplementary angles.
Q
Another illustration:
m R + m  P = 180.
150º
R
P
30º
 R and  P are supplementary
angles.
 R is a supplement to  P .
 P is a supplement to  R.
Chapter 2:
Angle Relationships
Topic: Angle Pairs
(continuation)
Objectives
Enumerate the different kinds of
angle pairs
Define each kind of angle pair;
Identify & Illustrate the different
kinds of angle pairs, clearly
showing their relations.
Introduction
Relationships exist between angles.
If two angles have the same
measure, then they are
CONGRUENT.
For example, if mA = 50 and mB
= 50, then A  B .
By the sum of their measures,
relations can be established.
Introduction
Relationship is very
important.
- relationship to your fellow
students.
- relationship to our friends,
neighbors, & other people.
Introduction
Relationship to our
family.
Relationship to God.
We MUST learn to
value relationships.
As the saying goes,
“ No man is an
island. No man can
stand alone”
Look at this figure…
Consider RPD and  QPD.
- share a common vertex(P),
- Share a common side
(segment PD)
- but no interior points in
common.
 RPD and  QPD are
Adjacent angles.
R
.S
D
.A
P
Q
ADJACENT ANGLES
Are angles meeting at a
common vertex (corner)
and sharing a common
side but NO interior
points in common.
Consider this figure
 RPD and  QPD
are Adjacent
angles &
complementary.
R
.S
D
.A
P
Q
How about the other pairs of angles in
the figure?
Like ,
 RPD and  QPR ?
 QPD and  QPR ?
Are these pairs of angles
Adjacent or not ? why?
These pairs of angle are
NON – ADJACENT
ANGLES.
R
.S
D
.A
P
Q
Consider this figure
B
70º
Q
20º
Can we say that  B and  Q are
Complementary?
Adjacent or non-adjacent?
 B and  Q are Complementary angles
BUT non- adjacent angles.
Another illustration:
150º
R
P
30º
m R + m  P = 180.
 R and  P are supplementary angles
and non - adjacent angles.
Consider the following:
D
145º
R
35º
P
Q
 RPD and  QPD are
supplementary angles and Adjacent
angles.
What Consider
can you say
ray PR &
the about
following:
ray PQ of RPD & QPD?
D
145º
R
35º
P
Q
They are non- common sides & opposite
rays.
 RPD and  QPD are LINEAR PAIR of
angles.
Definition of LINEAR PAIR
Are TWO adjacent angles and whose
non common sides are opposite
rays.
LINEAR PAIR POSTULATE
States that “ Linear pair of angles are
supplementary”
In the figure:
D
145º
R
P
35º
Q
 RPD and  QPD are LINEAR PAIR
of angles and supplementary.
In the figure, name & identify
linear pair of angles.
D
A
P
B
C
APC and BPC, APD and APC
APD and DPB, DPD and BPC
are LINEAR PAIR of angles.
REMEMBER THIS…..
LINEAR PAIR of angles
are adjacent and
supplementary.
In the figure, we can write an
equation. Like,
D
A
P
C
mAPC +mBPC = 180
mAPD + mAPC = 180
mAPD + mDPB = 180
mDPD + mBPC = 180
B
In the figure, if mAPD = 120. . What
is the measure of the other angles?
D
A
P
C
mAPC +mBPC = 180
mAPD + mAPC = 180
mAPD + mDPB = 180
mDPD + mBPC = 180
B
In the figure, if mAPD = 120. . What
is the measure of the other angles?
A
C
120°
60°
D
60°
P
120°
B
mAPD + mAPC = 180 (linear pair postulate)
120 + mAPC = 180 ( by substitution)
mAPC = 60( by subtraction)
In the given figure, what are nonadjacent angles?
A
C
120°
60°
D
60°
P
120°
B
APD and BPC
APC and BPD
These non-adjacent angles are also called
vertical angles.
Vertical Angles
D
A
P
B
C
 In the figure, APC and BPD, APD
and BPC are vertical angles.
Vertical Angles
ARE TWO NON
ADJACENT ANGLES
formed by two
intersecting lines.
APC and BPD,
APD and BPC are NON
ADJACENT angles.
Line AB and line CD
are two intersecting
lines
D
A
P
C
B
What can you say about the
measures of the vertical angles?
A
C
120°
60°
D
60°
P
120°
B
APD
mAPDand
= mBPC
BPC
APC
mAPC
and
=m
BPD
BPD
These non-adjacent angles are also called
vertical angles.
Vertical Angles Theorem
Vertical angles are
congruent.
Fixing skills
•
In the given
figure,
APB and CPD are
right angles.
Name all pairs of:
1. Complementary
angles.
A
B
1
2
3 4
P
5 6
C
7 8
ANSWERS:
3 AND 4
5 AND 6
Fixing skills
•
In the given
figure,
APB and CPD are
right angles.
Name all pairs of:
2. Supplementary
angles.
A
B
1
2
3 4
P
5 6
C
7 8
ANSWERS:
1 AND 2
7 AND 8
Fixing skills
•
In the given
figure,
APB and CPD are
right angles.
Name all pairs of:
3.
Vertically
opposite angles.
A
B
1
2
3 4
P
5 6
C
7 8
ANSWERS:
CPD and BPA
3 AND 6
APC and DPB
5 AND 4
Fixing skills
•
In the given
figure,
APB and CPD are
right angles.
Name all pairs of:
4. Linear pair of
angles.
A
B
1
2
3 4
P
5 6
C
7 8
ANSWERS:
1 AND 2
7 AND 8
Fixing skills
•
In the given
figure,
APB and CPD are
right angles.
Name all pairs of:
5.Adjacentangles.
ANSWERS:
1 AND 2
3 AND 4
5 AND 6
7 AND 8
A
B
1
2
3 4
P
5 6
C
7 8
STUDENT ACTIVITY
Define the following
pairs of angles:
• Adjacent angles
• Linear pair of angles
• Vertical angles
State the following:
• Linear pair postulate
• “Linear pair of angles are
supplementary”
• Vertical angles Theorem
• “Vertical angles are
congruent”
QUIZ
Get ¼ sheet of
paper
State whether each of the
following is TRUE or FALSE.
1. TWO ADJACENT RIGHT
ANGLES ARE
SUPPLEMENTARY.
2. ALL SUPPLEMENTARY
ANGLES ARE ADJACENT.
3. SOME SUPPLEMENTARY
ANGLES ARE LINEAR PAIR.
State whether each of the
following is TRUE or FALSE.
4. TWO VERTICAL
ANGLES ARE ALWAYS
CONGRUENT.
5.
ALL RIGHT ANGLES
ARE CONGRUENT.
ASSIGNMENT
• A. Copy & answer the
following. Show your
solution .
• ¼ sheet of paper
Find the value of x.
1.
4x – 20
2x + 10
Find the value of x.
2.
x + 35
2x - 5
ASSIGNMENT
•
•
•
•
B. DEFINE THE FOLLOWING:
- PERPENDICULAR LINES
- PERPENDICULAR BISECTOR
- EXTERIOR ANGLE OF A
TRIANGLE
• NOTE:
• Write your answer in your
notebook
B. Solve the following
• The supplement of a
certain angle is four
times larger than its
complement. What is
the measure of the
angle?
Find the value of x.
3.
4x
5x
x
Find the value of x.
4..
3x-5
2x + 10
Find the value of x.
B. Solve the following
• Two complementary
angles are on the ratio
1 : 4. What is the
measure of the larger
angle?
Solution
• Let x = the measure of the smaller
angle
• 4x = the measure of the larger
angle
• X + 4x = 90
• 5x = 90
• X = 18
• Therefore, the measure of the
larger angle is 4 ( 18 ) or 72.
SOLVE AND CHECK
• The measure of
one of two
complementary
angles is 15 less
than twice the
measure of the
other. Find the
measure of each
angle.
Solution:
Let x = measure
of the first
angle
2x – 15 = measure
of the second
angle.
Solution:
X + 2x – 15 = 90 (def. of complementary angles)
3x = 90 + 15
3x = 105
x = 35 ( measure of the 1st )
2x -15 = 55 (measure of the 2nd )
• The measure of one of two
complementary angles is 15 less than
twice the measure of the other. Find
the measure of each angle.
Check:
55 is 15 less than twice 35. the
sum of
35 and 55 is 90.