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Transcript
Geometry Lesson 2 – 8 Proving Angle Relationships Objective: Write proofs involving supplementary angles. Write proofs involving congruent and right angles. Postulate 2.10 Protractor Postulate Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180. Postulate 2.11 Angle Addition Postulate Use Angle Addition Postulate Find m1if m2 56 and mJKL 145. m1 m2 mJKL m1 56 145 m1 89 Example If m1 23 and mABC 131, find the measure of 3. Justify each step. m1 m2 m3 mABC 23 90 m3 131 113 m3 131 113 m3 113 131 113 m3 18 Angle Add. Post. Sub Sub Subt. Prop. Sub Theorems Supplement Theorem If two angles form a linear pair, then they are supplementary angles. Complement Theorem If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. Example Angles 6 & 7 form a linear pair. If m6 3x 32 & m7 5 x 12, find x, m6, & m7. Justify each step. m6 m7 180 Supplement Thm. 3x + 32 + 5x + 12 = 180 Sub 8x + 44 = 180 8x + 44 - 44 = 180 - 44 8x = 136 m6 83 m7 97 8 x 136 8 8 x = 17 Sub Subt. Prop. Sub Division Prop. Sub Properties of Angle Congruence Reflexive 1 1 Symmetric If 1 2, then 2 1 Transitive If 1 2 & 2 3, then 1 3. Theorem Congruent Supplement Theorem Angles supplementary to the same angle or to congruent angles are congruent. Abbreviation: Theorem Congruent Complements Theorem Angles complementary to the same angle or congruent angles are congruent. Abbreviation Prove that the vertical angles 2 and 4 are congruent. Given: 2 & 4 are vertical angles Prove: 2 4 Theorem 2.8 Vertical Angle Theorem If two angles are vertical angles, then they are congruent. Prove that if DB bisects ADC , then 2 3. Right Angle Theorems Theorem 2.9 Perpendicular lines intersect to form 4 right angles Theorem 2.10 All right angles are congruent. Theorem 2.11 Perpendicular lines from 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. Homework Pg. 154 1 – 4 all, 6, 8 – 14, 45 - 48 all