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NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 2-8 Study Guide and Intervention Proving Angle Relationships Supplementary and Complementary Angles There are two basic postulates for working with angles. The Protractor Postulate assigns numbers to angle measures, and the Angle Addition Postulate relates parts of an angle to the whole angle. Angle Addition Postulate R is in the interior of ∠PQS if and only if m∠PQR + m∠RQS = m∠PQS. The two postulates can be used to prove the following two theorems. Supplement Theorem If two angles form a linear pair, then they are supplementary angles. Example: If ∠1 and ∠2 form a linear pair, then m∠1 + m∠2 = 180. Complement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. ⃡ , then m∠3 + m∠4 = 90. Example: If ⃡𝐺𝐹 ⊥ 𝐺𝐻 Example 1: If ∠1 and ∠2 form a linear pair Example 2: If ∠1 and ∠2 form a right angle and m∠2 = 115, find m∠1. m∠1 + m∠2 = 180 m∠1 + 115 = 180 - 115 -115 m∠1 = 65 and m∠2 = 20, find m∠1. m∠1 + m∠2 = 90 m∠1 + 20 = 90 Supplement Theorem Substitution Subtraction Prop. Substitution - 20 -20 m∠1 = 70 Complement Theorem Substitution Subtraction Prop. Substitution Congruent and Right Angles The Reflexive Property of Congruence, Symmetric Property of Congruence, and Transitive Property of Congruence all hold true for angles. The following theorems also hold true for angles. Congruent Supplements Theorem Angles supplement to the same angle or congruent angles are congruent. Congruent Compliments Theorem Angles compliment to the same angle or to congruent angles are congruent. Vertical Angles Theorem If two angles are vertical angles, then they are congruent. Theorem 2.9 Perpendicular lines intersect to form four right angles. Theorem 2.10 All right angles are congruent. Theorem 2.11 Perpendicular lines form congruent adjacent angles. Theorem 2.12 If two angles are congruent and supplementary, then each angle is a right angle. Theorem 2.13 If two congruent angles form a linear pair, then they are right angles. Chapter 2 49 Glencoe Geometry NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 2-8 Practice Proving Angle Relationships Find the measure of each numbered angle and name the theorems that justify your work. 1. m∠ 1 = x + 10 m∠ 2 = 3x + 18 2. m∠ 4 = 2x – 5 m∠ 5 = 4x – 13 3. m∠ 6 = 7x – 24 m∠ 7 = 5x + 14 4. Write a two-column proof. Given: ∠ 1 and ∠ 2 form a linear pair. ∠ 2 and ∠ 3 are supplementary. Prove: ∠ 1 ≅ ∠ 3 Chapter 2 49 Glencoe Geometry