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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. STATEMENT REASONS What you know Postulated, definitions, theorems, properties, etc.

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**Prove the Vertical Angles Theorem**

Theorem 2-1: Vertical Angles Theorem Vertical angles are congruent. Given: 1 and 2 are vertical angles Prove: 1 2 STATEMENT REASON 1 and 2 are vertical angles Def. of vertical angles m1 + m3 = 180 Angle Addition Postulate m2 + m3 = 180 Angle Addition Postulate m1 + m3 = m2 + m3 Substitution m1 + m3 - m3 = m2 + m3 - m3 Subtraction prop. of Equality (SPE) m1 = m2 Simplify 1 2 Vertical Angle Theorem

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**Use Theorem 2-1 (Vertical Angle Theorem) **

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

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**The vertical angles, as we found, measure 107º**

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

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**Prove the Congruence Supplements Theorem **

(Vertical Angle Thm is a special case of this Thm) Theorem 2-2: Congruence Supplement Theorem If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. 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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**Prove Theorem 2-2: Congruence Supplement Theorem**

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 3 STATEMENT REASON 1 and 2 are supplementary 3 and 2 are supplementary Given Def of Sup. s m1 + m2 = 180 m3 + m2 = 180 m1 + m2 = m3 + m2 Substitution m1 + m2 - m2 = m3 + m2 - m2 Subtraction prop. of Equality (SPE) m1 = m3 Simplify 1 3 Congruence Supplement Theorem

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**Prove the Congruence Complements Theorem**

Do Now Prove the Congruence Complements Theorem Theorem 2-3 If two angles are complements of the same angle (or of congruent angles), then the two angles are congruent. Given: 1 and 2 are complementary 3 and 2 are complementary Prove: 1 3 STATEMENT REASON 1 and 2 are complementary 3 and 2 are complementary Given m = + m2 = 90 m Def of Comp s m3 Substitution m1 + m2 = m3 + m2 m1 + m2 - m2 = m3 + m2 - m2 SPE Simplify 1 3

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**Prove Theorem 2-2: Congruence Supplement Theorem**

If two angles are supplements of the same angle (or of congruent angles), then the two angles are congruent. Given: 1 and 2 are supplementary 3 and 2 are supplementary Prove: 1 3 STATEMENT REASON 1 and 2 are supplementary 3 and 2 are supplementary Given Def of Sup. s m1 + m2 = 180 m3 + m2 = 180 m1 + m2 = m3 + m2 Substitution m1 + m2 - m2 = m3 + m2 - m2 Subtraction prop. of Equality (SPE) m1 = m3 Simplify 1 3 Congruence Supplement Theorem

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3.2 Proof and Perpendicular Lines

3.2 Proof and Perpendicular Lines

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