1. 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

2. 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

3. 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.

4. 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.

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

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

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

8. 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.

9. 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°

10. 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°

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

12. EXAMPLE 4
Standardized Test Practice
SOLUTION
Because TPQ and QPR form a linear pair, the sum of their measures is 180.
The correct answer is B.
ANSWER

13. Use the diagram in Example 4.
GUIDED PRACTICE
for Example 4
Use the diagram in Example 4.
7. Solve for x.
SOLUTION
Because TPQ and QPR form a linear pair, the sum of their measures is 180°.
The correct answer is B.
32 + (3x +1) = 180
Original equation
32 + 3x +1 = 180
Distributive property of equality
3x = 147
Subtract 33 from each side
x = 49
Divide each side by 3

14. GUIDED PRACTICE
for Example 4
Use the diagram in Example 4.
8. Find m TPS.
SOLUTION
m TPS = (3x + 1)°
Substitute the value x = 49
m TPS = ( )°
m TPS = (147 +1)°
m TPS = 148°