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Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio 45202 Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240 Lesson 2-1 Inductive Reasoning and Conjecture Lesson 2-2 Logic Lesson 2-3 Conditional Statements Lesson 2-4 Deductive Reasoning Lesson 2-5 Postulates and Paragraph Proofs Lesson 2-6 Algebraic Proof Lesson 2-7 Proving Segment Relationships Lesson 2-8 Proving Angle Relationships Example 3 Use Supplementary Angles Example 4 Vertical Angles OBJECTIVE: To write proofs involving congruent and right angles (2.9.8J) (M8.D.1.1) THEOREMS Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive. • REFLEXIVE: 1 1 • SYMMETRIC: If 1 2, then 2 1 • TRANSITIVE: If 1 2 , and 2 3 then 1 3 , THEOREMS • 2.6 Angles supplementary to the same angle or to congruent angles are congruent. • 2.7 Angles complementary to the same angle or to congruent angles are congruent. • 2.8 Vertical Angle Theorem: If two angles are vertical angles, then they are congruent. THEOREMS • 2.9 Perpendicular lines intersect to form 4 right angles. • 2.10 All right angles are congruent. • 2.11 Perpendicular lines form congruent adjacent angles. • 2.12 If 2 angles are congruent and supplementary, then each angles is a right angle. • 2.13 If 2 congruent angles for a linear pair, then they are right angles. In the figure, Given: Prove: and form a linear pair, and and Prove that are congruent. form a linear pair. Proof: Statements Reasons 1. 1. Given 2. 2. Linear pairs are supplementary. 3. Definition of supplementary angles 3. 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Definition of congruent angles In the figure, NYR and RYA form a linear pair, AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXY are congruent. Proof: Statements Reasons 1. 1. Given linear pairs. 2. 2. If two s form a linear pair, then they are suppl. s. 3. 3. Given 4. 4. If 1 and 2 are vertical angles and m1 m2 find m1 and m2. 1 m1 2 Vertical Angles Theorem m2 Definition of congruent angles Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3. and Answer: m1 = 37 and m2 = 37 If and are vertical angles and find and Answer: mA = 52; mZ = 52 and REVIEW 1. Two angles that are nonadjacent are (always, sometimes, or never) vertical? SOMETIMES 2. Two angles that are congruent are (always, sometimes, or never) complementary to the same angle? SOMETIMES REVIEW 3. The measures of 2 complementary angles are in the ratio 4:1. What is the measure of the smaller angle? A. 15 B. 18 C. 24 D. 36 B 4. T is the set of all positive numbers n such that n < 50 and square root of n is an integer. What is the median of the members of set T? A. 4 B. 16 C. 20 D. 25 B End of Custom Shows WARNING! Do Not Remove This slide is intentionally blank and is set to auto-advance to end custom shows and return to the main presentation.