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2-6. Geometric Proof. Warm Up. Lesson Presentation. Lesson Quiz. Holt McDougal Geometry. Holt Geometry. Objectives : Write two-column proofs. Prove geometric theorems by using deductive reasoning (uses facts, rules, definitions to make conjectures from given situations) Vocabulary :

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2-6

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  1. 2-6 Geometric Proof Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

  2. Objectives: Write two-column proofs. Prove geometric theorems by using deductive reasoning (uses facts, rules, definitions to make conjectures from given situations) Vocabulary: Theorem – a statement that can be proven true. Two-column proof – format for proofs where the statements are listed on the left and the reasons are listed on the right.

  3. Definitions • Postulates • Properties • Theorems Conclusion Hypothesis When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.

  4. Example 1: Writing Justifications Write a justification for each step, given that A and Bare supplementary and mA = 45°. 1. A and Bare supplementary. mA = 45° Given information Def. of supp s 2. mA + mB= 180° Subst. Prop of = 3. 45° + mB= 180° Steps 1, 2 Subtr. Prop of = 4. mB= 135°

  5. 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. Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you.

  6. Check It Out! 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: • 1 and 2 are supp., and 2 and 3 are supp. b. m1 + m2 = m2 + m3 c. Subtr. Prop. of = d. 1  3

  7. 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 m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary.

  8. Example 3 Continued Given 1 and 2 are supplementary. 1  3 m1 + m2 = 180° Def. of supp. s m1 = m3 Def. of s Subst. m3 + m2 = 180° Def. of supp. s 3 and 2 are supplementary

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