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Objectives: Prove and apply theorems about angles

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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Objectives: Prove and apply theorems about angles

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  1. 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

  2. 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.

  3. 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 STATEMENTREASON 1 and 2 are vertical angles Def. of vertical angles Angle Addition Postulate m1 + m3 = 180 Angle Addition Postulate m2 + m3 = 180 Substitution m1 + m3 = m2 + m3 Subtraction prop. of Equality (SPE) m1 + m3 - m3 = m2 + m3 - m3 Simplify m1 = m2 Vertical Angle Theorem 1  2

  4. 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. ProblemReason 4x – 101 = 2x + 3 Vertical Angles Theorem 4x = 2x + 104 Addition Property of Equality 2x = 104 Subtraction Property of Equality x = 52 Division Property of Equality

  5. The vertical angles, as we found, measure 107º 107 107 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

  6. 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

  7. 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 STATEMENTREASON 1 and 2 are supplementary 3 and 2 are supplementary Given Def of Sup. s m1 + m2 = 180 m3 + m2 = 180 Substitution m1 + m2 = m3 + m2 Subtraction prop. of Equality (SPE) m1 + m2 - m2 = m3 + m2 - m2 Simplify m1 = m3 Congruence Supplement Theorem 1  3

  8. 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 STATEMENTREASON 1 and 2 are complementary 3 and 2 are complementary Given m2 90 m1 + = + m2 = 90 Def of Comp s m3 Substitution m1 + m2 = m3 + m2 SPE m1 + m2 - m2 = m3 + m2 - m2 Simplify 1  3

  9. 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 STATEMENTREASON 1 and 2 are supplementary 3 and 2 are supplementary Given Def of Sup. s m1 + m2 = 180 m3 + m2 = 180 Substitution m1 + m2 = m3 + m2 Subtraction prop. of Equality (SPE) m1 + m2 - m2 = m3 + m2 - m2 Simplify m1 = m3 Congruence Supplement Theorem 1  3

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