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Geometry Proofs Obj: To Complete Proofs Involving Segment Theorems To complete proofs involving angle theorems 5 Essential Parts to a Good Proof State the theorem to be proved List the given information If possible draw a diagram to illustrate the given information State what is to be proved Develop a system of deductive reasoning Deductive Reasoning Definitions & postulates-Statements accepted as true w/o proof Theorem- undefined terms that can be used as truth once they have been proven Theorem: Congruence of segments is reflexive, symmetric, and transitive Written as symbols: Reflexive: AB AB Symmetric: IF AB CD, then CD AB Transitive: IF AB CD and CD EF then AB EF Example: D C Given: B ABCD Prove: AD AB BC CD Statements A Justification 1.) ABCD Given 1.) _________ 2.) AD = AB + BD Seg. + post 2.) ___________ 3.) BD = BC + CD Seg. + post. 3.)___________ 4.) AD = AB + BC + CD substitution 4.)___________ Your Turn: P Q R Given: pts PQRS are collinear S Prove: PQ = PS - QS Statement 1.) pts PQRS are collinear 2.) PS = PQ + QS 3.) PS – QS = PQ 4.) PQ = PS - QS Justification Solution to “Your Turn”: Given: pts PQRS are collinear Prove: PQ = PS - QS Statement Justification 1.) pts PQRS are collinear 1.) Given 2.) PS = PQ + QS 2.) Seg. + post 3.) PS – QS = PQ 3.) Subtraction 4.) PQ = PS - QS 4.) symmetric Prop. Verifying Angle Relationships Theorems Supplement thm-if 2 angles form a linear pr then they are supplementary Congruence of angles is reflexive, symmetric, and transitive Angles supplementary to the same angle or to congruent angles are congruent Theorems Continued Angles complementary to the same angle or congruent angles are congruent All right angles are congruent Vertical angles are congruent Perpendicular lines intersect to form 4 right angles Example: If A and C are vertical ' s, and mA 3x 2 and mC 2 x 4 . Find mA& mC. Angle A = Angle C, Vert. angles 3x - 2 = 2x + 4 -2x +2 -2x +2 x=6 A = 3(6) - 2 = 16o C = 2 (6) + 4 = 16o Your Turn: Find the measure of each numbered angle m1 2 x 5 m2 x 4 2x - 5 + x - 4 = 180 3x - 9 = 180 3x = 189 x = 63 1 2 m1 2 (63) - 5 = 121o m2 63 - 4 = 59o Homework: Put this in your agenda: Pg 116 5 – 12, 21 – 24 Pg 128 9 – 21 odd, 22 - 28