Give a reason for each statement. 1. If m 1 = 90º and m 2 = 90º, then m 1 = m 2. 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)

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

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.

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)

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

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

Prove the Vertical Angles Congruence Theorem EXAMPLE 3 Prove the Vertical Angles Congruence Theorem GIVEN: 5 and 7 are vertical angles. PROVE: ∠ 5 ∠ 7 STATEMENT REASONS 5 and 7 are vertical angles. 1. 1. Given 2. 5 and 6 are a linear pair. 6 and 7 are a linear pair. 2. Definition of linear pair, as shown in the diagram 3. 5 and 6 are supplementary. 6 and 7 are supplementary. 3. Linear Pair Postulate 4. ∠ 5 ∠ 7 Congruent Supplements Theorem 4.

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

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

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