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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 Contents

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) Lesson 8 Contents

THEOREMS Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive. REFLEXIVE: SYMMETRIC: If , then TRANSITIVE: If , and , then

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, form a linear pair, and Prove that are congruent. and Given: form a linear pair. Prove: Example 8-3a

2. Linear pairs are supplementary. 2. Proof: Statements Reasons 1. Given 1. 2. Linear pairs are supplementary. 2. 3. Definition of supplementary angles 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Definition of congruent angles 6. Example 8-3b

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. Example 8-3c

2. If two s form a linear pair, then they are suppl. s. Proof: Statements Reasons 1. Given 2. If two s form a linear pair, then they are suppl. s. 3. Given 4. 1. 2. 3. linear pairs. Example 8-3d

If 1 and 2 are vertical angles and m1 and m2 find m1 and m2. Vertical Angles Theorem 1 2 Definition of congruent angles m1 m2 Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3. Example 8-4a

Answer: m1 = 37 and m2 = 37 Example 8-4b

If and are vertical angles and and find and If and are vertical angles and and Answer: mA = 52; mZ = 52 Example 8-4b

REVIEW Two angles that are nonadjacent are (always, sometimes, or never) vertical? 2. Two angles that are congruent are (always, sometimes, or never) complementary to the same angle? SOMETIMES SOMETIMES

REVIEW 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 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 B

End of Lesson 8

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