2.6 Proving Statements about Angles

1. 2.6 Proving Statements about Angles

2. Properties of Angle Congruence Reflexive For any angle, A <A <A. Symmetric If <A <B, then <B <A. Transitive If <A <B and <B <C, then <A <C.

3. Right Angle Congruence Theorem • All right angles are congruent. . A . X . . Y B Z C

4. Congruent Supplements Theorem • If two angles are supplementary to the same angle, then they are congruent • If m<1 + m<2 = 180° and m<2 + m<3 = 180°, then m<1 = m<3 or

5. Congruent Complements Theorem • If two angles are complementary to the same angle, then the two angles are congruent. • If m<4 + m<5 = 90° and m<5 + m<6 = 90°, then m<4 = m<6 or

6. Linear Pair Postulate • If two angles form a linear pair, then they are supplementary. 1 2 m<1 + m<2 = 180°

7. Example: • < 1 and < 2 are a linear pair. If m<1 = 78°, then find m<2.

8. Vertical Angles Theorem • Vertical angles are congruent. 1 4 2 3

9. Example <1 and <2 are complementary angles. <1 and <3 are vertical angles. If m<3 = 49°, find m<2.

10. Proving the Right Angle Congruence Theorem Given: Angle 1 and angle 2 are right angles Prove: Statements Reasons 1. Given 2. Def. of right ’s 3. Trans. POE 4. Def. of  ’s

11. Proving the Vertical Angles Theorem 5 7 6 Given: 5 and 6 are a linear pair. 6 and 7 are a linear pair. Prove: 5  7 Statements Reasons • 5 and 6 are a linear pair. 6 and 7 are a linear pair. 1. Given 2. 5 and 6 are supplementary.6 and 7 are supplementary. Linear Pair Postulate 3.  Supplements Theorem

12. Solve for x.

13. Give a reason for each step of the proof. Choose from the list of reasons given.

14. Given: 6  7 Prove: 5  8 Plan for Proof: First show that 5  6 and 7  8. Then use transitivity to show that 5  8.) Statements Reasons 1. 6  7 1. Given 7  8 2. Vertical ’s Theorem 3. 6  8 3. Trans. POC 4. 5  6 4. Vertical ’s Theorem 5. Trans. POC 5. 5  8