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C HAPTER flow charts and paragraph proof

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SAT P ROBLEM OF THE DAY

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SAT P ROBLEM OF THE DAY SOLUTION Right Answer: D

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O BJECTIVES Write flowchart and paragraph proofs. Prove geometric theorems by using deductive reasoning.

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F LOW CHART A second style of proof is a flowchart proof, which uses boxes and arrows to show the structure of the proof. The justification for each step is written below the box.

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T HEOREM

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E XAMPLE #1 Use the given flowchart proof to write a two- column proof. Given: 2 and 3 are comp. 1 3 Prove: 2 and 1 are comp.

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CONTINUE Flowchart proof:

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CONTINUE Two-column proof:

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EXAMPLE #2 Use the given flowchart proof to write a two- column proof. Given: RS = UV, ST = TU Prove: RT TV

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CONTINUE Flowchart proof:

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C ONTINUE Two-column proof:

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E XAMPLE #3 Use the given two-column proof to write a flowchart proof. Given: B is the midpoint of AC. Prove: 2 AB = AC

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SOLUTION Flowchart proof:

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S TUDENT GUIDED PRACTICE Do problems 7 and 8 from your book page 123

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P ARAGRAPH PROOFS What are paragraph proofs? A paragraph proof is a style of proof that presents the steps of the proof and their matching reasons as sentences in a paragraph. Although this style of proof is less formal than a two-column proof, you still must include every step.

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T HEOREMS

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E XAMPLE Use the given paragraph proof to write a two- column proof. Given: m 1 + m 2 = m 4 Prove: m 3 + m 1 + m 2 = 180° Paragraph Proof: It is given that m1 + m2 = m4. 3 and 4 are supplementary by the Linear Pair Theorem. So m3 + m4 = 180° by definition. By Substitution, m3 + m1 + m2 = 180°.

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S OLUTION

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E XAMPLE #2 Use the given paragraph proof to write a two- column proof. Given: WXY is a right angle. 1 3 Prove: 1 and 2 are complementary. Paragraph Proof: Since WXY is a right angle, mWXY = 90° by the definition of a right angle. By the Angle Addition Postulate, mWXY = m2 + m3. By substitution, m2 + m3 = 90°. Since 1 3, m1 = m3 by the definition of congruent angles. Using substitution, m2 + m1 = 90°. Thus by the definition of complementary angles, 1 and 2 are complementary.

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S OLUTION Statements Reasons 1. WXY is a right angle. 1. Given 2. m WXY = 90° 2. Def. of right angle 3. m 2 + m 3 = m WXY 3. Angle Add. Postulate 4. m 2 + m 3 = 90° 4. Subst. 5. 1 3 5. Given 6. m 1 = m 3 6. Def. of s 7. m 2 + m 1 = 90° 7. Subst. 8. 1 and 2 are comp. 8. Def. of comp. angles

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E XAMPLE Use the given two-column proof to write a paragraph proof. Given: 1 and 2 are complementary Prove: 3 and 4 are complementary m 3 + m 4 = 90 ° 3 and 4 are comp.

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S OLUTION Paragraph:

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S TUDENT GUIDED P RACTICE Do problems in the book page 123 and124

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H OMEWORK !! Do problems 3-5 on you book page 122

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C LOSURE Today we learned about flowcharts, two column paragraphs and how to use them to make proofs. Next class, we are going to start with chapter 3.

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