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Flowchart and Paragraph Proofs

2-7 Flowchart and Paragraph Proofs Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

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Warm Up Complete each sentence.

1. If the measures of two angles are ? , then the angles are congruent. 2. If two angles form a ? , then they are supplementary. 3. If two angles are complementary to the same angle, then the two angles are ? . equal linear pair congruent

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Objectives Write 2-column proof, flowchart and paragraph proofs.

Prove geometric theorems by using deductive reasoning.

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Vocabulary flowchart proof 2-column proof paragraph proof

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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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Example 1: Reading a Flowchart Proof

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. Flowchart proof:

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Statements Reasons Example 1 Continued Two-column proof:

1. 2 and 3 are comp. 1 3 1. Given 2. m2 + m3 = 90° 2. Def. of comp. s 3. m1 = m3 3. Def. of s 4. m2 + m1 = 90° 4. Subst. 5. 2 and 1 are comp. 5. Def. of comp. s

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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 Flowchart proof:

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Check It Out! Example 2 Continued

Statements Reasons 1. RS = UV, ST = TU 1. Given 2. RS + ST = TU + UV 2. Add. Prop. of = 3. RS + ST = RT, TU + UV = TV 3. Seg. Add. Post. 4. RT = TV 4. Subst. 5. RT TV 5. Def. of segs.

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Example 3: Writing a Flowchart Proof

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

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Example 3 Continued Flowchart proof:

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Check It Out! Example 4 Use the given two-column proof to write a flowchart proof. Given: 2 4 Prove: m1 m3 Two-column Proof:

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Check It Out! Example 4 Continued

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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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Example 5: Reading a Paragraph Proof

Use the given paragraph proof to write a two-column proof. Given: m1 + m2 = m4 Prove: m3 + m1 + m2 = 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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Statements Reasons Example 5 Continued Two-column proof:

1. m1 + m2 = m4 1. Given 2. 3 and 4 are supp. 2. Linear Pair Theorem 3. m3 + m4 = 180° 3. Def. of supp. s 4. m3 + m1 + m2 = 180° 4. Substitution

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Check It Out! Example 6 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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Check It Out! Example 6 Continued

Statements Reasons 1. WXY is a right angle. 1. Given 2. mWXY = 90° 2. Def. of right angle 3. m2 + m3 = mWXY 3. Angle Add. Postulate 4. m2 + m3 = 90° 4. Subst. 5. 1 3 5. Given 6. m1 = m3 6. Def. of s 7. m2 + m1 = 90° 7. Subst. 8. 1 and 2 are comp. 8. Def. of comp. angles

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Example 7: Writing a Paragraph Proof

Use the given two-column proof to write a paragraph proof. Given: 1 and 2 are complementary Prove: 3 and 4 are complementary m3 + m4 = 90° 3 and 4 are comp.

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Example 7 Continued Paragraph proof:

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Check It Out! Example 8 Use the given two-column proof to write a paragraph proof. Given: 1 4 Prove: 2 3 Two-column proof:

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Check It Out! Example 8 Continued

Paragraph proof: It is given that 1 4. By the Vertical Angles Theorem, 1 2 and 3 4. By the Transitive Property of Congruence, 2 4. Also by the Transitive Property of Congruence, 2 3.

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Lesson Quiz Use the two-column proof at right to write the following. 1. a flowchart proof 2. a paragraph proof

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