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EXAMPLE 1

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

tobias-gregory
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EXAMPLE 1

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  1. ABBC, DCBC GIVEN: BC PROVE: REASONS STATEMENT 1. 1. ABBC, DCBC Given 2. 2. Definition of perpendicular lines Band Care right angles. 3. BC 3. Right Angles Congruence Theorem EXAMPLE 1 Use right angle congruence Write a proof.

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

  3. REASONS STATEMENT 1. 1. 1 and 2 are supplements. Given 3 and 2 are supplements. 2. 2. m1+m2= 180° Definition of supplementary angles m3+m2= 180° 3. m3+m2 3. m1+m2= Transitive Property of Equality 4. m1=m3 4. Subtraction Property of Equality 5. • 3 5. Definition of congruent angles EXAMPLE 2 Prove a case of Congruent Supplements Theorem

  4. ANSWER 2 Steps for Examples 1 and 2 GUIDED PRACTICE 1. How many steps do you save in the proof in Example 1 by using the Right Angles Congruence Theorem? 2. Draw a diagram and write GIVEN and PROVE statements for a proof of each case of the Congruent Complements Theorem.

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

  6. Statements (Reasons) 1.1 and 3 are complements; 3 and 5 are complements. (Given) 2.∠ 1 5 (Congruent Complements Theorem.) for Examples 1 and 2 GUIDED PRACTICE

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

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

  9. 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° for Example 3 GUIDED PRACTICE

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

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

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

  13. ANSWER x = 49 for Example 4 GUIDED PRACTICE Use the diagram in Example 4. 7. Solve for x.

  14. 8. Find m TPS. m TPS = 148° ANSWER for Example 4 GUIDED PRACTICE Use the diagram in Example 4.

  15. 1. Give the reason for each step (Given) ANSWER (Def. of ) ANSWER (Def. oflinear pair ) ANSWER Daily Homework Quiz 1 5 GIVEN : 1 PROVE : is supplementary to 4 Statements (Reasons) 1. 1 5 2. m 1 = m 5 3. 4 and are a linear pair. 5

  16. (Def. ofsupplementary ) (Linear Pair Post.) (Def. ofsupplementary ) (Substitution Prop. of Eq.) ANSWER ANSWER ANSWER ANSWER Daily Homework Quiz 4. 4 and are supplementary . 5 5. m 4 + m 5 = 180 6. m 4 + m 1 = 180 7. 1 is supplementary to 4.

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