1. Give a reason for each statement.
1. If m = 90º and m = 90º, then m = m
ANSWER
(Transitive Prop. of Eq.)
2. If AB BC , then ABC is a right angle. ┴
ANSWER
(Def. of perpendicular)
3. If FG RS, then FG = RS =
ANSWER
(Def. of segment congruence)

2. Writing Proofs of Geometric Relationships
Target
Writing Proofs of Geometric Relationships
You will…
Use properties of special angle pairs.

3. Theorems
Right Angles Congruence Theorem 2.3 – All right angles are congruent. (If two angles are right angles, then they are congruent. Proof – page 116)
Congruent Supplements Theorem 2.4 – If two angles are supplementary to the same angle or congruent angles, then they are congruent. (Proof for one case – page 117)
Congruent Complements Theorem 2.5 – If two angles are complementary to the same angle or congruent angles, then they are congruent.

4. Theorems
Linear Pair Postulate 12 – If two angles form a linear pair, then they are supplementary.
Vertical Angles Congruence Theorem 2.6 – Vertical angles are congruent. ( If two angles are vertical, then they are congruent. Proof – page 118)

5. 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

6. Prove a case of Congruent Supplements Theorem
EXAMPLE 2
Prove a case of Congruent Supplements Theorem
GIVEN:
1 and are supplements.
3 and are supplements.
PROVE: 3
STATEMENT
REASONS 1.
3 and are supplements.
1 and are supplements.
Given 1. 2.
m m = 180°
m m = 180° 2.
Def. of supplementary angles 3.
m m =
m m 2
Transitive Prop. of = 3. 4.
m = m
Subtraction. Prop. of = 4. 5. 3
Def. of angles 5.

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

8. GUIDED PRACTICE
for Example 3
In Exercises 3–5, use the diagram.
3. If m = 112°, find m 2, m 3, and m
4. If m = 67°, find m 1, m 3, and m
5. If m = 71°, find m 1, m 2, and m
6. Which previously proven theorem is used here?
Congruent Supplements Theorem
ANSWER
ANSWER
m = 68°
m = 112°
m = 68°
ANSWER
m = 113°
m = 113°
m = 67°
ANSWER
m = 109°
m = 71°
m = 109°

9. 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

10. GUIDED PRACTICE
for Example 4
Use the diagram in Example 4.
7. Solve for x.
x = 49
ANSWER
8. Find m TPS.
m TPS = 148°
ANSWER