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NAME ______________________________________________ DATE 2-8 ____________ PERIOD _____ 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. Protractor Postulate and a number r between 0 and 180, there is exactly one ray Given AB , such that the measure with endpoint A, extending on either side of AB of the angle formed is r. Angle Addition Postulate R is in the interior of PQS if and only if mPQR mRQS mPQS. P R Q S The two postulates can be used to prove the following two theorems. If two angles form a linear pair, then they are supplementary angles. If 1 and 2 form a linear pair, then m1 m2 180. D Complement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. ⊥ GH , then m3 m4 90. If GF Example 1 B C F J 3 4 G H Example 2 If 1 and 2 form a linear pair and m2 115, find m1. If 1 and 2 form a right angle and m2 20, find m1. R Q W 2 2 1 M 2 1 A N 1 S P m1 m2 180 m1 115 180 m1 65 T m1 m2 90 m1 20 90 m1 70 Suppl. Theorem Substitution Subtraction Prop. Compl. Theorem Substitution Subtraction Prop. Exercises Find the measure of each numbered angle. 1. 2. T P 7 Q 8 8 R S m7 5x 5, m8 x 5 © Y X Glencoe/McGraw-Hill U 7 6 Z 5 V W m5 5x, m6 4x 6, m7 10x, m8 12x 12 99 3. A 13 F 11 12 H J C m11 11x, m12 10x 10 Glencoe Geometry Lesson 2-8 Supplement Theorem NAME ______________________________________________ DATE 2-8 ____________ PERIOD _____ Study Guide and Intervention (continued) Proving Angle Relationships Congruent and Right Angles Three properties of angles can be proved as theorems. Congruence of angles is reflexive, symmetric, and transitive. Angles supplementary to the same angle or to congruent angles are congruent. Angles complementary to the same angle or to congruent angles are congruent. C K 1 F B A 2 D E H G T N 4 3 M J L Q If 1 and 2 are supplementary to 3, then 1 2. R 6 5 P S If 4 and 5 are complementary to 6, then 4 5. Example Write a two-column proof. Given: ABC and CBD are complementary. DBE and CBD form a right angle. Prove: ABC DBE Statements 1. ABC and CBD are complementary. DBE and CBD form a right angle. 2. DBE and CBD are complementary. 3. ABC DBE C A D B E Reasons 1. Given 2. Complement Theorem 3. Angles complementary to the same are . Exercises Complete each proof. 1. Given: A B ⊥B C ; 1 and 3 are complementary. Prove: 2 3 © A 1 2 B 3 2. Given: 1 and 2 form a linear pair. m1 m3 180 Prove: 2 3 E C D T L 1 2 B 3 R P Statements Reasons Statements Reasons a. A B ⊥B C b. a. b. Definition of ⊥ a. 1 and 2 form a linear pair. m1 m3 180 a. Given c. m1 m2 mABC d. 1 and 2 form a rt . e. 1 and 2 are compl. f. c. b. b. Suppl. Theorem d. c. 1 is suppl. to 3. c. e. d. d. suppl. to the same are . f. Given g. 2 3 g. Glencoe/McGraw-Hill 100 Glencoe Geometry NAME ______________________________________________ DATE 2-8 ____________ PERIOD _____ Skills Practice Proving Angle Relationships Find the measure of each numbered angle. 2. m5 22 1 2 4. m13 4x 11, m14 3x 1 3. m1 38 5 2 1 6 5. 9 and 10 are complementary. 7 9, m8 41 6. m2 4x 26, m3 3x 4 2 13 14 7 8 9 10 3 Determine whether the following statements are always, sometimes, or never true. 7. Two angles that are supplementary form a linear pair. 8. Two angles that are vertical are adjacent. 9. Copy and complete the following proof. Given: QPS TPR Prove: QPR TPS Proof: R Q Statements Reasons a. a. b. mQPS mTPR b. c. mQPS mQPR mRPS c. S T P mTPR mTPS mRPS © d. d. Substitution e. e. f. f. Glencoe/McGraw-Hill 101 Glencoe Geometry Lesson 2-8 1. m2 57 NAME ______________________________________________ DATE 2-8 ____________ PERIOD _____ Practice Proving Angle Relationships Find the measure of each numbered angle. 1. m1 x 10 m2 3x 18 2. m4 2x 5 m5 4x 13 3. m6 7x 24 m7 5x 14 4 3 5 1 6 7 2 Determine whether the following statements are always, sometimes, or never true. 4. Two angles that are supplementary are complementary. 5. Complementary angles are congruent. 6. Write a two-column proof. Given: 1 and 2 form a linear pair. 2 and 3 are supplementary. Prove: 1 3 1 2 3 7. STREETS Refer to the figure. Barton Road and Olive Tree Lane form a right angle at their intersection. Tryon Street forms a 57° angle with Olive Tree Lane. What is the measure of the acute angle Tryon Street forms with Barton Road? © Glencoe/McGraw-Hill 102 Barton Rd Tryon St Olive Tree Lane Glencoe Geometry NAME ______________________________________________ DATE 2-8 ____________ PERIOD _____ Reading to Learn Mathematics Proving Angle Relationships Pre-Activity How do scissors illustrate supplementary angles? Read the introduction to Lesson 2-8 at the top of page 107 in your textbook. Is it possible to open a pair of scissors so that the angles formed by the two blades, a blade and a handle, and the two handles, are all congruent? If so, explain how this could happen. Reading the Lesson 1. Complete each sentence to form a statement that is always true. a. If two angles form a linear pair, then they are adjacent and . . c. If D is a point in the interior of ABC, then mABC mABD and a number x between d. Given RS and , there is exactly one ray with endpoint R, extended on either side of RS, such that the measure of the angle formed is x. . Lesson 2-8 b. If two angles are complementary to the same angle, then they are e. If two angles are congruent and supplementary, then each angle is a(n) angle. f. lines form congruent adjacent angles. g. “Every angle is congruent to itself” is a statement of the of angle congruence. Property h. If two congruent angles form a linear pair, then the measure of each angle is . i. If the noncommon sides of two adjacent angles form a right angle, then the angles are . 2. Determine whether each statement is always, sometimes, or never true. a. Supplementary angles are congruent. b. If two angles form a linear pair, they are complementary. c. Two vertical angles are supplementary. d. Two adjacent angles form a linear pair. e. Two vertical angles form a linear pair. f. Complementary angles are congruent. g. Two angles that are congruent to the same angle are congruent to each other. h. Complementary angles are adjacent angles. Helping You Remember 3. A good way to remember something is to explain it to someone else. Suppose that a classmate thinks that two angles can only be vertical angles if one angle lies above the other. How can you explain to him the meaning of vertical angles, using the word vertex in your explanation? © Glencoe/McGraw-Hill 103 Glencoe Geometry