Special Pairs of Angles
2-4
EXAMPLE 1
Identify complements and supplements
In the figure, name a pair of
complementary angles, a pair
of supplementary angles, and
a pair of adjacent angles.
SOLUTION
Because 32°+ 58° = 90°, BAC and
complementary angles.
Because 122° + 58° = 180°,
supplementary angles.
RST are
CAD and
RST are
Because BAC and CAD share a common vertex and
side, they are adjacent.
GUIDED PRACTICE
1.
for Example 1
In the figure, name a pair of complementary
angles, a pair of supplementary angles, and a
pair of adjacent angles.
Because 41° + 49° = 90°, FGK
and GKL are complementary
angles.
Because 49° + 131° = 180°,
supplementary angles.
HGK and
GKL are
Because FGK and HGK share a common vertex and
side, they are adjacent.
GUIDED PRACTICE
2.
for Example 1
Are KGH and LKG adjacent angles ? Are
FGK and FGH adjacent angles? Explain.
KGH and LKG do not share a common vertex , they
are not adjacent.
FGK and FGH have common interior points, they
are not adjacent.
EXAMPLE 2 Find measures of a complement and a supplement
a.
Given that
find m 2.
1 is a complement of
2 and m
1 = 68°,
SOLUTION
a.
You can draw a diagram with complementary
adjacent angles to illustrate the relationship.
m
2 = 90° – m 1 = 90° – 68° = 22
EXAMPLE 2 Find measures of a complement and a supplement
b.
Given that
find m 3.
3 is a supplement of
4 and m
4 = 56°,
SOLUTION
b.
You can draw a diagram with supplementary
adjacent angles to illustrate the relationship.
m
3 = 180° – m 4 = 180° –56° = 124°
EXAMPLE 3 Find angle measures
Sports
When viewed from the side, the frame of a ballreturn net forms a pair of supplementary angles with
the ground. Find m BCE and m ECD.
EXAMPLE 3 Find angle measures
SOLUTION
STEP 1
m
Use the fact that the sum of the measures
of supplementary angles is 180°.
BCE + m  ECD = 180° Write equation.
(4x+ 8)° + (x + 2)° = 180°
5x + 10 = 180
5x = 170
x = 34
Substitute.
Combine like terms.
Subtract 10 from each side.
Divide each side by 5.
EXAMPLE 3 Find angle measures
STEP 2
Evaluate: the original expressions when x = 34.
m
BCE = (4x + 8)° = (4 34 + 8)° = 144°
m
ANSWER
ECD = (x + 2)° = ( 34 + 2)° = 36°
The angle measures are 144° and 36°.
GUIDED PRACTICE
3.
Given that
find m 1.
for Examples 2 and 3
1 is a complement of
2 and m
SOLUTION
You can draw a diagram with complementary
adjacent angle to illustrate the relationship
1
m
2 8°
1 = 90° – m
2 = 90°– 8° = 82°
2 = 8° ,
for Examples 2 and 3
GUIDED PRACTICE
Given that
find m 4.
4.
3 is a supplement of
4 and m
SOLUTION
You can draw a diagram with supplementary
adjacent angle to illustrate the relationship
m
4 = 180° – m
3 = 180°– 117° = 63°
117°
3 4
3 = 117°,
GUIDED PRACTICE
for Examples 2 and 3
LMN and PQR are complementary angles.
Find the measures of the angles if m LMN =
(4x – 2)° and m PQR = (9x + 1)°.
5.
SOLUTION
m
LMN + m
PQR = 90°
(4x – 2 )° + ( 9x + 1 )° = 90°
13x – 1 = 90
13x = 91
x =7
Complementary angle
Substitute value
Combine like terms
Add 1 to each side
Divide 13 from each side
GUIDED PRACTICE
for Examples 2 and 3
Evaluate the original expression when x = 7
m
LMN = (4x – 2 )° = (4·7 – 2 )° = 26°
m
PQR = (9x – 1 )° = (9·7 + 1)° = 64°
ANSWER
m
LMN = 26°
m
PQR = 64°
EXAMPLE 4
Identify angle pairs
Identify all of the linear pairs and all of the
vertical angles in the figure at the right.
SOLUTION
To find vertical angles, look or
angles formed by intersecting lines.
ANSWER
1 and
5 are vertical angles.
To find linear pairs, look for adjacent angles whose
noncommon sides are opposite rays.
ANSWER
1 and 4 are a linear pair.
are also a linear pair.
4 and
5
EXAMPLE 5
Find angle measures in a linear pair
ALGEBRA
Two angles form a linear pair. The measure of
one angle is 5 times the measure of the other.
Find the measure of each angle.
SOLUTION
Let x° be the measure of one
angle. The measure of the
other angle is 5x°. Then use
the fact that the angles of a
linear pair are supplementary
to write an equation.
EXAMPLE 5
x + 5x = 180°
6x = 180°
x = 30°
ANSWER
Find angle measures in a linear pair
Write an equation.
Combine like terms.
Divide each side by 6.
The measures of the angles are 30°
and 5(30)° = 150°.
GUIDED PRACTICE
6.
For Examples 4 and 5
Do any of the numbered angles in the diagram
below form a linear pair?Which angles are vertical
angles? Explain.
ANSWER
No, adjacent angles have their non common
sides as opposite rays, 1 and 4 , 2 and
3 and 6, these pairs of angles have sides
that from two pairs of opposite rays
5,
GUIDED PRACTICE
For Examples 4 and 5
The measure of an angle is twice the measure of
its complement. Find the measure of each angle.
7.
SOLUTION
Let x° be the measure of one angle . The
measure of the other angle is 2x° then
use the fact that the angles and their
complement are complementary to write
an equation
x° + 2x° = 90° Write an equation
3x = 90 Combine like terms
x = 30 Divide each side by 3
ANSWER
The measure of the angles are 30° and
2( 30 )° = 60°
EXAMPLE 2
Name the property shown
Name the property illustrated by the statement.
a. If
T and
R
b. If NK
T
BD , then BD
P, then
R
P.
NK .
SOLUTION
a. Transitive Property of Angle Congruence
b. Symmetric Property of Segment Congruence
GUIDED PRACTICE
2.
CD
for Example 2
CD
ANSWER
Reflexive Property of Congruence
3.
If
Q
V, then
V
Q.
ANSWER
Symmetric Property of Congruence
EXAMPLE 3
Prove the Vertical Angles Congruence Theore
Prove vertical angles are congruent.
GIVEN:
5 and
7 are vertical angles.
PROVE:  5  7
EXAMPLE 3
Prove the Vertical Angles Congruence Theore
STATEMENT
REASONS
1.
5 and
7 are vertical angles. 1.Given
2.
5 and
6 and
6 are a linear pair.
7 are a linear pair.
5 and
6 and
6 are supplementary. 3.Linear Pair Postulate
7 are supplementary.
3.
4.  5  7
2.Definition of linear pair,
as shown in the
diagram
4.Congruent
Supplements Theorem
GUIDED PRACTICE
for Example 3
In Exercises 3–5, use the diagram.
3.
If m
1 = 112°, find m
ANSWER
m
2 = 68°
m
3 = 112°
m
4 = 68°
2, m
3, and m
4.
GUIDED PRACTICE
4.
If m
for Example 3
2 = 67°, find m
1, m
3, and m
4.
1, m
2, and m
3.
ANSWER
m
1 = 113°
5.
m
3 = 113°
m
4 = 67°
If m
4 = 71°, find m
ANSWER
m
1 = 109°
m
2 = 71°
m
3 = 109°
GUIDED PRACTICE
6.
for Example 3
Which previously proven theorem is used in
Example 3 as a reason?
ANSWER
Congruent Supplements Theorem
EXAMPLE 1
Use right angle congruence
Write a proof.
AB
GIVEN:
B
PROVE:
BC , DC
C
REASONS
STATEMENT
1. AB
2.
3.
BC , DC
B and
angles.
B
BC
BC
C are right
1. Given
2. Definition of perpendicular
lines
C
3.Right Angles Congruence
Theorem
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
Prove that two angles supplementary to the same angle
are congruent.
GIVEN: 1 and 2 are supplements.
3 and 2 are supplements.
PROVE:
1
3
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
REASONS
STATEMENT
1.
1 and
3 and
2 are supplements. 1. Given
2 are supplements.
2. m 1+ m 2 = 180°
m 3+ m 2 = 180°
1+m
3. m
2= m
2. Definition of
supplementary angles
3+m
2 3. Transitive Property of
Equality
4. m
1=m
3
4. Subtraction
Property of Equality
5.
1
3
5. Definition of
congruent angles
GUIDED PRACTICE
for Examples 1 and 2
1. How many steps do you save in the proof in
Example 1 by using the Right Angles Congruence
Theorem?
ANSWER
2 Steps
2.
Draw a diagram and write GIVEN and PROVE
statements for a proof of each case of the
Congruent Complements Theorem.
GUIDED PRACTICE
for Examples 1 and 2
ANSWER
Write a proof.
Given:
1 and
3 and
Prove:  1
3 are complements;
5 are complements.
5
GUIDED PRACTICE
for Examples 1 and 2
Statements (Reasons)
1.
1 and
3 and
3 are complements;
5 are complements.
(Given)
2.  1
5 Congruent Complements Theorem.