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Section 2-5: Proving Angles Congruent SPI 32E: solve problems involving complementary, supplementary, congruent, vertical or adjacent angles given angle measures expressed algebraically. Objectives: Prove and apply theorems about angles

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Using Deductive Reasoning to show a Conjecture is True Deductive Reasoning (logical) Reason from a given statement to produce a conclusion Real-world examples: Doctors diagnose a patients illness Carpenters to determine what materials are needed for a job. Proof Set of steps you take to show a conjecture is true Theorem The statement that you prove to be true Format of a Proof to derive a Theorem (Side by Side) Given: What you know Prove: What you will show to be true, based on known information. STATEMENTREASONS What you knowPostulated, definitions, theorems, properties, etc.

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Prove the Vertical Angles Theorem Given: 1 and 2 are vertical angles Prove: 1 2 STATEMENTREASON 1 and 2 are vertical angles m 1 + m 3 = 180 m 2 + m 3 = 180 m 1 + m 3 = m 2 + m 3 m 1 + m 3 - m 3 = m 2 + m 3 - m 3 m 1 = m Vertical angles are congruent. Theorem 2-1: Vertical Angles Theorem Def. of vertical angles Angle Addition Postulate Substitution Subtraction prop. of Equality (SPE) Simplify Vertical Angle Theorem

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The angles with labeled measures are vertical angles because their sides are opposite rays. Apply the Vertical Angles Theorem to find x. Find the value of x. Problem Reason 4x – 101=2x + 3 Vertical Angles Theorem 4x=2x Addition Property of Equality 2x=104 Subtraction Property of Equality x=52 Division Property of Equality Use Theorem 2-1 (Vertical Angle Theorem) to solve problems since it is proven.

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The vertical angles, as we found, measure 107º 107 What is the measure of the other pair of vertical angles? How do you know? (What Def, postulate, theorem….?) Def: Adjacent angles are supplementary and vertical angles are congruent 73 º HELP!!

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Prove the Congruence Supplements Theorem (Vertical Angle Thm is a special case of this Thm) If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2-2: Congruence Supplement Theorem What do you know based on the definition of: Supplementary Angles? Two angles whose measures have a sum of 180 Congruent Angles? Two angles have the same measure

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Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 3 STATEMENTREASON 1 and 2 are supplementary 3 and 2 are supplementary m 1 + m 2 = 180 m 3 + m 2 = 180 m 1 + m 2 = m 3 + m 2 m 1 + m 2 - m 2 = m 3 + m 2 - m 2 m 1 = m Given Def of Sup. s Substitution Subtraction prop. of Equality (SPE) Simplify Congruence Supplement Theorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Prove Theorem 2-2: Congruence Supplement Theorem

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If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Theorem 2-3 Prove the Congruence Complements Theorem Given: 1 and 2 are complementary 3 and 2 are complementary Prove: 1 3 STATEMENTREASON 1 and 2 are complementary 3 and 2 are complementary m 1 + = + m 2 = 90 Def of Comp s Substitution m 1 + m 2 - m 2 = m 3 + m 2 - m 2 Simplify Given m 2 90 m 3 m 1 + m 2 = m 3 + m 2 SPE 1 3

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Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 3 STATEMENTREASON 1 and 2 are supplementary 3 and 2 are supplementary m 1 + m 2 = 180 m 3 + m 2 = 180 m 1 + m 2 = m 3 + m 2 m 1 + m 2 - m 2 = m 3 + m 2 - m 2 m 1 = m Given Def of Sup. s Substitution Subtraction prop. of Equality (SPE) Simplify Congruence Supplement Theorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Prove Theorem 2-2: Congruence Supplement Theorem

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