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Geometric Proof Warm Up Determine whether each statement is true or false. If false, give a counterexample. 1. It two angles are complementary, then they are not congruent. false; 45° and 45° 2. If two angles are congruent to the same angle, then they are congruent to each other. true 3. Supplementary angles are congruent. false; 60° and 120° Holt McDougal Geometry Geometric Proof Essential Question How do we write two-column proofs and prove geometric theorems by using deductive reasoning? Unit 2Day 8 Geometric Proof 2-3 Holt McDougal Geometry Geometric Proof A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture. In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column. Holt McDougal Geometry Geometric Proof Example 1: Completing a Two-Column Proof Fill in the blanks to complete the two-column proof. Given: XY Prove: XY XY Statements 1. XY 2. XY = XY 3. XY . XY Holt McDougal Geometry Reasons 1. Given 2. Reflex. . Prop. of = 3. Def. of segs. Geometric Proof Example 2 Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem. Given: 1 and 2 are supplementary, and 2 and 3 are supplementary. Prove: 1 3 Proof: a. 1 and 2 are supp., and 2 and 3 are supp. b. m1 + m2 = m2 + m3 c. Subtr. Prop. of = d. 1 3 Holt McDougal Geometry Geometric Proof Holt McDougal Geometry Geometric Proof Example 3: Writing a Two-Column Proof from a Plan Use the given plan to write a two-column proof. Given: 1 and 2 are supplementary, and 1 3 Prove: 3 and 2 are supplementary. Plan: Use the definitions of supplementary and congruent angles and substitution to show that m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary. Holt McDougal Geometry Geometric Proof Example 3 Continued Statements 1. 1 and 2 are supplementary. Reasons 1. Given 1 3 2. m1 + m2 = 180° of supp. s 2. Def. . = m3 3. m1 . 3. Def. of s 4. m3 + m2 = 180° 4. Subst. 5. 3 and 2 are supplementary 5. Def. of supp. s Holt McDougal Geometry Geometric Proof Example 4 Use the given plan to write a two-column proof if one case of Congruent Complements Theorem. Given: 1 and 2 are complementary, and 2 and 3 are complementary. Prove: 1 3 Plan: The measures of complementary angles add to 90° by definition. Use substitution to show that the sums of both pairs are equal. Use the Subtraction Property and the definition of congruent angles to conclude that 1 3. Holt McDougal Geometry Geometric Proof Check It Out! Example 3 Continued Statements Reasons 1. 1 and 2 are complementary. 1. Given 2 and 3 are complementary. 2. m1 + m2 = 90° m2 + m3 = 90° of comp. s 2. Def. . + m2 = m2 + m3 3. Subst. 3. m1 . 4. m2 = m2 4. Reflex. Prop. of = 5. m1 = m3 5. Subtr. Prop. of = 6. 1 3 6. Def. of s Holt McDougal Geometry Geometric Proof Example 5 Use the given plan to write a two-column proof. Given: 1, 2 , 3, 4 Prove: m1 + m2 = m1 + m4 Plan: Use the linear Pair Theorem to show that the angle pairs are supplementary. Then use the definition of supplementary and substitution. Statements 1. 1 and 2 are supp. Reasons 1. Linear pair thm. 1 and 4 are supp. 2. m1 + m2 = 180°, 2. Def. of supp. s m1 + m4 = 180° 3. m1 + m2 = m1 + m4 Holt McDougal Geometry 3. Subst. Geometric Proof Assignment: • p. 61 # 1-4 • p. 62 # 6,7 Holt McDougal Geometry Geometric Proof Holt McDougal Geometry Geometric Proof Holt McDougal Geometry