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Honors Geometry Lesson 2.6 Deductive Reasoning What You Should Learn Why You Should Learn It Goal 1: How to identify a special angle relationships Goal 2: How to use deductive reasoning to verify angle relationships Deductive reasoning allows you to build a mathematical system and to solve real-life problems Identifying Special Pairs of Angles Vertical Angles Two angles are vertical angles if their sides form two pairs of opposite rays 1 2 Identifying Special Pairs of Angles Linear pair Two adjacent angles are a linear pair if their noncommon sides are on opposite rays 1 2 Identifying Special Pairs of Angles Complementary Angles Two angles are complementary if the sum of their measures is 90°. Each angle is the complement of the other. 60° 30° Identifying Special Pairs of Angles Supplementary Angles Two angles are supplementary if the sum of their measures is 180°. Each angle is the supplement of the other. 80° 100° Example 1 Use the terms vertical angles, linear pair, complementary angles & supplementary angles to describe the relationships between the labeled angles in the figure 3 6 4 5 Example 1 Solution Vertical Angles: 3 and 5, 4 and 6 Linear Pairs: 3 and 4, 4 and 5, 5 and 6, 6 and 3 Complementary Angles: None Supplementary Angles: 3 and 4, 4 and 5, 5 and 6, 6 and 3 3 6 4 5 Postulate 11: Linear Pair Postulate If two angles form a linear pair, then they are supplementary, i.e., the sum of their measures is 180° Deductive Reasoning To deduce means to reason from known facts When you prove a theorem, you are using deductive reasoning using the existing structure to deduce new parts of the structure In geometry, as in construction, new parts of the structure are built upon existing parts Theorem 2.1: Congruent Supplements Theorem If two angles are supplementary to the same angle or to congruent angles, then they are congruent Proof of Theorem 2.1: Congruent Supplements Theorem Given: 1 is the supplement of 2 3 is the supplement of 4 2 4 Prove: 1 3 m1 + m2 = 180 Def. of Supplementary Angles m3 + m4 = 180 Def. of Supplementary Angles m1 + m2 = m3 + m4 Transitive Prop. of = 2 4 Given m2 = m4 Def. of Congruence m1 + m2 = m3 + m2 Subsitution Prop. of = m1 = m3 Subtraction Property of Equality 1 3 Definition of Congruence Theorem 2.2 Congruent Complements Theorem If two angles are complementary to the same angle or to congruent angles, then they are congruent Proof of Theorem 2.2 Congruent Complements Theorem Fill in the missing steps and missing reasons Proof of Theorem 2.2 (Solution) Congruent Complements Theorem Theorem 2.3: Vertical Angles Theorem If two angles are vertical angles, then they are congruent.