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2.6 Proving Statements about Angles 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. Right Angle Congruence Theorem • All right angles are congruent. A . X B .C Y . .Z 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 1 3 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 4 6 Linear Pair Postulate • If two angles form a linear pair, then they are supplementary. 1 2 m<1 + m<2 = 180° Example: • < 1 and < 2 are a linear pair. If m<1 = 78°, then find m<2. Vertical Angles Theorem • Vertical angles are congruent. 1 4 2 3 1 3 , 2 4 Example <1 and <2 are complementary angles. <1 and <3 are vertical angles. If m<3 = 49°, find m<2. Proving the Right Angle Congruence Theorem Given: Angle 1 and angle 2 are right angles 1 2 Prove: Statements Reasons 1 . 1 and 2 right ' s 1. Given 2 . 1 90 and 2 90 2. Def. of right ’s 3 . m 1 m 2 3. Trans. POE 4. 1 2 4. Def. of ’s Proving the Vertical Angles Theorem 5 6 7 Given: 5 and 6 are a linear pair. 6 and 7 are a linear pair. Prove: 5 7 Statements Reasons 1. 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. 2. Linear Pair Postulate 3. 5 7 3. Supplements Theorem Solve for x. Give a reason for each step of the proof. Choose from the list of reasons given. 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 2. 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