2.8 Proving angle relationships cont. ink.notebook. September 20, page 82 page cont. page 83. page 84. Standards. Cont.

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1 2.8 Proving angle relationships cont. ink.notebook page 82 page cont. page 83 page 84 Lesson Objectives Standards Lesson Notes 2.8 Proving Angle Relationships Cont. Press the tabs to view details. 1

2 Lesson Objectives Standards Lesson Notes Lesson Objectives Standards Lesson Notes After this lesson, you should be able to successfully write proofs involving complementary and supplementary angles, congruent angles, and right angles. G.CO.9 Prove theorems about lines and angles. Press the tabs to view details. Press the tabs to view details. page 55 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. Theorem 2.10: Right Angle Congruence Theorem: All right angles are. 2

3 Given: 1 and 2 are right angles Prove: 1 2 STATEMENTS REASONS 1. 1 and 2 are right angles 1. GIVEN 2. m 1 = 90; m 2=90 2. DEF. OF RT. ANGLE 3. m 1 = m 2 3. SUBSTITUTION PROP DEF OF CONGRUENT ANGLES pull rectangles away to reveal answers Supplement Theorem: (Linear Pair Theorem) If two angles form a linear pair, then they are. 1 and 2 form a linear pair, so 1 and 2 are supplementary and m 1 + m 2 =. 3

4 Theorem 2.6: Congruent Supplements Theorem: If two angles are supplementary to the same angle (or to congruent angles), then they are. If 1 and 2 are supplementary and 3 and 2 are supplementary, then. Theorem 2.7: Congruent Complements Theorem: If two angles are complementary to the same angle (or to congruent angles), then they are. If 4 and 5 are complementary and 6 and 5 are complementary, then. 4

5 2.8 Proving angle relationships cont. ink.notebook Theorem 2.8: Vertical Angles Congruence Theorem: Vertical Angles are. Other Right Angle Theorems: 2.9: Perpendicular lines intersect to form right angles. 2.11: Perpendicular lines form congruent angles. 2.12: If two angles are and, then they are right angles. 2.13: If two congruent angles form a linear pair, then they are angles. 5

6 2.8 Proving angle relationships cont. ink.notebook STATEMENTS REASONS 1. GIVEN 2. m 1 = 90 m 2=90 2. DEF OF PERPENDICULAR LINES ALL RT. ANGLES CONGRUENT 2 STATEMENTS REASONS 1. 1 and 2 are supplements 1. GIVEN 1 and 4 are supplements Congruent Supp. THM 3. m 2 = m 4 3. DEF. OF CONGRUENT ANGLES 4. m 2 = GIVEN 5. m 4 = Substitution 6

7 STATEMENTS REASONS 1. 4 is a right angle 1. GIVEN 2. m 4 = DEF. OF RT. ANGLE VERTICAL ANGLES CONGRUENT THM 4. m 2 = m 4 4. DEF. OF CONGRUENT ANGLES 5. m 2 =90 5. SUBSTITUTION 6. m 2 + m 4 = ADDITION 7. 2 AND 4 are Supp. 7. Def of SUPP ANGLE A E G B C If m 1 + m 2 = 90 then 1 is complementary to 2 If m 1 + m 2 = m 3 + m 2 then m 1 = m 3 If AC º EG then ABG is a right angle If m 1 = 55 and m 3 = 55 then m 1 = m 3 If m 1 = m 2 then 1 2 If 3 4 then m 3 = m 4 If 1 and 3 are vertical angles then 1 3 Complementa Angles Subtractio Property of Equality Perpendicular Substitutio Property Congruent An Congruent An Theorem: Vertical Ang are Congrue M A B D A B N O C D A B E C A B C MN + NO = MO If ABC CBD then BC bisects ABD m ABD + m DBC = m ABC If m ABE is a right angle then m ABE = 90 If 1 is supplementary to 2 then m 1 + m 2 = 180 If 3 4 and 4 5 then 3 5 If B is the midpoint of AC then AB BC Segment Addi Postulate Angle Bisect Angle Additi Postulate Right Angle Supplementa Angles Transitive Property Midpoint 7

8 On the worksheet Practice 1. a.) If m 4 = 63, find m 1 and m Write and solve an equation to find x. Use x to find m AEB. b.) If m 3 = 121, find m 1, m 2, and m 4. 8

9 4. ple 3. 6:Given: 1 and 4 form a linear pair and m 1 + m 3 = 180 Prove: 3 4 g. 5. Given: 3 and 2 are complementary m 1 + m 2 = 90 Prove: Given: AC = BD Prove: AB = CD 9

10 B 8. Match the correct reason for each step in the proof Given: m 1 = m 2 Prove: m RSU = m TSV 1 R T E O F S 2 U V Answers: 1a) m 1 = 117, m 2 = 63 1b) m 1 = 121, m 2 = 59, m 4 = and 4 form a linear pair given, 1 and 4 are supp supp thm, m 1 + m 3 = 180 given, 1 and 3 are supp supp thm, 3 4 supp thm Book Work Pg #6, 9, 10, 12, given, def of comp s, given, substitution, subtraction, substitution 7. given, def of bisect, segment add post, substitution, addition 10

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