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

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: 3

Prove a case of Congruent Supplements Theorem EXAMPLE 2 Prove a case of Congruent Supplements Theorem 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. Definition of supplementary angles 3. m 1 + m 2 = m 3 + m 2 Transitive Property of Equality 3. 4. m 1 = m 3 Subtraction Property of Equality 4. 5. 3 Definition of congruent angles 5.

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 Write a proof. Given: 1 and 3 are complements; 3 and 5 are complements. Prove: ∠ 1 5 ANSWER

GUIDED PRACTICE for Examples 1 and 2 Statements (Reasons) 1. 1 and 3 are complements; 3 and 5 are complements. (Given) 2. ∠ 1 5 Congruent Complements Theorem.

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

Prove the Vertical Angles Congruence Theorem EXAMPLE 3 Prove the Vertical Angles Congruence Theorem STATEMENT REASONS 5 and 7 are vertical angles. 1. 1. Given 2. 5 and 7 are a linear pair. 6 and 7 are a linear pair. 2. Definition of linear pair, as shown in the diagram 3. 5 and 7 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. ANSWER m 2 = 68° m 3 = 112° m 4 = 68°

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

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

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

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

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 = (3 49 +1)° m TPS = (147 +1)° m TPS = 148°