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Geometry 2-8 - Proving Angle Relationships A. Supplementary and Complementary Angles 1. Postulate 2-11 - Angle Addition Postulate - If R is in the interior of ∠PQS , then m∠PQR + m∠ _____ = m∠PQS . The converse is also true: If m∠PQR + m∠RQS = m∠PQS , then R is in the interior of ∠PQS . Example 1: If m∠PQR = 2 x + 12 , m∠RQS = 3x − 2 , and m∠PQS = 50, find the value of x and m∠PQR . P Q R S P Q R S 2. Theorem 2-3 – Supplemental Theorem - If two angles form a linear pair, then they are ________________. 3. Theorem 2-4 - Complement Theorem - If the noncommon sides of two adjacent angles form a right angle, then the angles are complimentary angles. Example 2: If ∠1 and ∠2 form a linear pair, and m ∠1 = 4 x − 5 and m ∠2 = 14 x + 5 , find x, and m ∠1 . 2 1 4. Theorem 2-5 - Congruence of angles is reflexive, symmetric, and transitive a. Reflexive Property - ∠1 ≅ ∠1 b. Symmetric Property - If ∠1 ≅ ∠2 , then_______. c. Transitive Property - If ∠1 ≅ ∠2 , and ∠2 ≅ ∠3 , then ∠1 ≅ ∠3 . 5. Theorem 2-6 - Angles supplementary to the same angle or to congruent angles are congruent. a. If m∠1 + m∠2 = 180 , and m∠2 + m∠3 = 180 , then ___________ 2 1 3 Example 3: Proof of Theorem 2-6 Given: ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary Prove: ∠1 ≅ ∠3 Proof: Statements 1. ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary 2. m∠1 + m∠2 = 180 m∠2 + m∠3 = 180 3. 4. m∠1 = m∠3 5. ∠1 ≅ ∠3 6. Theorem 2-7 - Angles complementary To the same angle or to congruent angles are congruent. a. If m∠1 + m∠2 = 90 , and m∠2 + m∠3 = 90 , then __________. 3 2 1 Reasons 1. Given 2. 3. Substitution 4. 5. 1 2 3 7. Theorem 2-8 - If two angles are vertical angles, then they are congruent. Example 4: If ∠1 and ∠2 are vertical angles, and m∠1 = d − 32 , and m∠2 = 175 − 2d , find m∠1 and m∠2 . 8. Theorem 2-9 - Perpendicular lines intersect to form four _______________. 9. Theorem 2-10 - All right angles are congruent. 10. Theorem 2-11 - Perpendicular lines form congruent adjacent angles. 11. Theorem 2-12 - If 2 angles are congruent and supplementary, then each angles is a right angle. 12. If two congruent angles form a linear pair, then they are _____________. HW: Geometry 2-8 p. 112-114 16-32 even, 39, 44-45, 49-55 odd