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Congruence Theorems for Right and Vertical Angles

Learn and apply the Right Angle Congruence Theorem and the Vertical Angles Congruence Theorem to prove the congruence of angles in geometric problems.

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Congruence Theorems for Right and Vertical Angles

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  1. REASONS STATEMENT EXAMPLE 1 Use right angle congruence Theorem: Right Angle Congruence Theorem All right angles are congruent Given: <1 and <2 are right angles Prove: <1 is congruent to <2

  2. 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.

  3. 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

  4. 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

  5. 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.

  6. 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

  7. 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

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

  9. 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

  10. 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

  11. 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

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

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

  14. 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

  15. 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

  16. 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

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

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