1. Special Pairs of Angles
2-4

2. 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 RST are complementary angles.
Because 122° + 58° = 180°, CAD and RST are supplementary angles.
Because BAC and CAD share a common vertex and side, they are adjacent.

3. GUIDED PRACTICE
for Example 1
In the figure, name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles. 1.
Because 41° + 49° = 90°, FGK and GKL are complementary angles.
Because 49° + 131° = 180°, HGK and GKL are supplementary angles.
Because FGK and HGK share a common vertex and side, they are adjacent.

4. GUIDED PRACTICE
for Example 1
Are KGH and LKG adjacent angles ? Are FGK and FGH adjacent angles? Explain. 2.
KGH and LKG do not share a common vertex , they are not adjacent.
FGK and FGH have common interior points, they are not adjacent.

5. EXAMPLE 2
Find measures of a complement and a supplement
Given that 1 is a complement of and m 1 = 68°,
find m 2.
SOLUTION
a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.
m = 90° – m 1 = 90° – 68° = 22

6. EXAMPLE 2
Find measures of a complement and a supplement
b. Given that is a supplement of 4 and m = 56°,
find m 3.
SOLUTION
b. You can draw a diagram with supplementary adjacent angles to illustrate the relationship.
m 3 = 180° – m 4 = 180° –56° = 124°

7. EXAMPLE 3
Find angle measures
Sports
When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.

8. EXAMPLE 3 Find angle measures SOLUTION STEP 1
Use the fact that the sum of the measures of supplementary angles is 180°.
m BCE + m ∠ ECD = 180°
Write equation.
(4x+ 8)° + (x + 2)° = 180°
Substitute.
5x + 10 = 180
Combine like terms.
5x = 170
Subtract 10 from each side.
x = 34
Divide each side by 5.

9. EXAMPLE 3
Find angle measures
STEP 2
Evaluate: the original expressions when x = 34.
m BCE = (4x + 8)° = ( )° = 144°
m ECD = (x + 2)° = ( )° = 36°
The angle measures are 144° and 36°.
ANSWER

10. 3. Given that 1 is a complement of 2 and m 2 = 8° , find m 1.
GUIDED PRACTICE
for Examples 2 and 3
3. Given that 1 is a complement of 2 and m 2 = 8° , find m 1.
SOLUTION
You can draw a diagram with complementary adjacent angle to illustrate the relationship 1 2 8°
m 1 = 90° – m = 90°– 8° = 82°

11. 4. Given that 3 is a supplement of 4 and m 3 = 117°, find m 4.
GUIDED PRACTICE
for Examples 2 and 3
4. Given that 3 is a supplement of 4 and m 3 = 117°, find m 4.
SOLUTION
You can draw a diagram with supplementary adjacent angle to illustrate the relationship
m 4 = 180° – m = 180°– 117° = 63° 3 4
117°

12. 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)°.
SOLUTION
m LMN + m PQR = 90°
Complementary angle
(4x – 2 )° + ( 9x + 1 )° = 90°
Substitute value
13x – 1 = 90
Combine like terms
13x = 91
Add 1 to each side
x = 7
Divide 13 from each side

13. 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°

14. 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.
1 and are vertical angles.
ANSWER
To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.
1 and 4 are a linear pair and are also a linear pair.
ANSWER

15. EXAMPLE 5
Find angle measures in a linear pair
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.
ALGEBRA
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.

16. Find angle measures in a linear pair
EXAMPLE 5
Find angle measures in a linear pair
x + 5x = 180°
Write an equation.
6x = 180°
Combine like terms.
x = 30°
Divide each side by 6.
The measures of the angles are 30° and 5(30)° = 150°.
ANSWER

17. GUIDED PRACTICE
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. 6.
ANSWER
No, adjacent angles have their non common sides as opposite rays, and , and 5, and 6, these pairs of angles have sides that from two pairs of opposite rays

18. The measure of the angles are 30° and 2( 30 )° = 60°
GUIDED PRACTICE
For Examples 4 and 5
7. The measure of an angle is twice the measure of its complement. Find the measure of each angle.
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°

19. EXAMPLE 2
Name the property shown
Name the property illustrated by the statement. a.
If R T and T P, then R P. b.
If NK BD , then BD NK .
SOLUTION
Transitive Property of Angle Congruence a. b.
Symmetric Property of Segment Congruence

20. GUIDED PRACTICE
for Example 2
2. CD CD
Reflexive Property of Congruence
ANSWER
3. If Q V, then V Q.
Symmetric Property of Congruence
ANSWER

21. EXAMPLE 3
Prove the Vertical Angles Congruence Theorem
Prove vertical angles are congruent.
GIVEN:
5 and are vertical angles.
PROVE:
∠ 5 ∠

22. Prove the Vertical Angles Congruence Theorem
EXAMPLE 3
Prove the Vertical Angles Congruence Theorem
STATEMENT
REASONS
5 and are vertical angles. 1. 1.
Given 2.
5 and are a linear pair.
6 and are a linear pair. 2.
Definition of linear pair, as shown in the diagram 3.
5 and are supplementary.
6 and are supplementary. 3.
Linear Pair Postulate 4.
∠ 5 ∠
Congruent Supplements Theorem 4.

23. GUIDED PRACTICE
for Example 3
In Exercises 3–5, use the diagram.
3. If m = 112°, find m 2, m 3, and m
ANSWER
m = 68°
m = 112°
m = 68°

24. GUIDED PRACTICE
for Example 3
4. If m = 67°, find m 1, m 3, and m
ANSWER
m = 113°
m = 113°
m = 67°
5. If m = 71°, find m 1, m 2, and m
ANSWER
m = 109°
m = 71°
m = 109°

25. GUIDED PRACTICE
for Example 3
6. Which previously proven theorem is used in Example 3 as a reason?
Congruent Supplements Theorem
ANSWER

26. Use right angle congruence
EXAMPLE 1
Use right angle congruence
Write a proof.
GIVEN:
AB BC , DC BC
PROVE:
B C
STATEMENT
REASONS 1.
AB BC , DC BC 1.
Given 2.
B and C are right angles. 2.
Definition of perpendicular
lines 3.
B C 3.
Right Angles Congruence
Theorem

27. EXAMPLE 2
Prove a case of Congruent Supplements Theorem
Prove that two angles supplementary to the same angle are congruent.
GIVEN:
1 and are supplements.
3 and are supplements.
PROVE: 3

28. Prove a case of Congruent Supplements Theorem
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
STATEMENT
REASONS 1.
3 and are supplements.
1 and are supplements.
Given 1. 2.
m m 2 = 180°
m m 2 = 180° 2.
Definition of supplementary angles 3.
m m =
m m 2
Transitive Property of Equality 3. 4.
m = m
Subtraction Property of Equality 4. 5. 3
Definition of congruent angles 5.

29. 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.

30. GUIDED PRACTICE
for Examples 1 and 2
Write a proof.
Given: and are complements; 3 and are complements.
Prove: ∠
ANSWER

31. GUIDED PRACTICE
for Examples 1 and 2
Statements (Reasons)
and are complements; 3 and are complements.
(Given)
2. ∠
Congruent Complements Theorem.