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Geometry unit 5.5
This document contains geometry content on indirect proofs, two-column proofs, and properties of angles. It includes examples of writing indirect proofs, two-column proofs, and solving equations. It quizzes the reader with problems involving justifying steps in solving equations and identifying properties such as the transitive property of equality.
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Geometry unit 5.5
- 2. GEOMETRY 4-5 Usingindirect reasoning 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
- 3. Writing a Two-ColumnProof from a Plan Use the given plan to write a two-column proof. Given: Ð1 and Ð2 are supplementary, and Ð1 @ Ð3 Prove: Ð3 and Ð2 are supplementary. Plan: Use the definitions of supplementary and congruent angles and substitution to show that mÐ3 + mÐ2 = 180°. By the definition of supplementary angles, Ð3 and Ð2 are supplementary.
- 4. Writing a Two-ColumnProof : Continued Statements Reasons Ð1 and Ð2 are supplementary. Ð1 @ Ð3 1. 1. 2. 2. . 3. . 3. 4. 4. 5. 5. Given mÐ1 + mÐ2 = 180° Def. of supp. Ðs mÐ1 = mÐ3 mÐ3 + mÐ2 = 180° Ð3 and Ð2 are supplementary Def. of @ Ðs Subst. Def. of supp. Ðs
- 5. TEACH! Writing aTwo-Column Proof Use the given plan to write a two-column proof if one case of Congruent Complements Theorem. Given: Ð1 and Ð2 are complementary, and Ð2 and Ð3 are complementary. Prove: Ð1 @ Ð3 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 Ð1 @ Ð3.
- 6. TEACH! Continued StatementsReasons Ð1 and Ð2 are complementary. Ð2 and Ð3 are complementary. 1. 1. 2. 2. . 3. . 3. 4. 4. 5. 5. 6. 6. Given mÐ1 + mÐ2 = 90° mÐ2 + mÐ3 = 90° Def. of comp. Ðs mÐ1 + mÐ2 = mÐ2 + mÐ3 mÐ2 = mÐ2 mÐ1 = mÐ3 Subst. Reflex. Prop. of = Subtr. Prop. of = Ð1 @ Ð3 Def. of @ Ðs
- 7. Use indirect reasoningto prove: If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Given: the cost of two items is more than $50. Prove: At least one of the items costs more than $25. Begin by assuming that the opposite is true. That is assume that neither item costs more than $25.
- 8. Use indirect reasoningto prove: If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Given: the cost of two items is more than $50. Prove: At least one of the items costs more Btheagnin $ b2y5 .assuming that the opposite is true. That is assume that neither item costs more tThhains m$2e5a.ns that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. So the assumption that neither items cost more than $25 must be incorrect.
- 9. Use indirect reasoningto prove: If Jacky spends more than $50 to buy two items at a bicycle shop, then at least one of the items costs more than $25. Therefore, at least one of the items costs more than $25. This means that both items cost $25 or less. This means that the two items together cost $50 or less. This contradicts the given information that the amount spent is more than $50. So the assumption that neither items cost more than $25 must be incorrect.
- 10. Writing an indirectproof Step-1: Assume that the opposite of what you want to prove is true. Step-2: Use logical reasoning to reach a contradiction to the earlier statement, such as the given information or a theorem. Then state that the assumption you made was false. Step-3: State that what you wanted to prove must be true
- 11. Write an indirectproof: LMN LMN D D DLMN Given: Prove: has at most one right angle. Indirect proof: Assume has more than one right angle. That is assume Ð L a n d Ð M are both right angles.
- 12. Write an indirectproof: LMN LMN D D Given: Prove: has at most one right angle. ÐL and ÐM If are both right angles, then mÐL=mÐM =90o According to the Triangle Angle Sum Theorem,. mÐL+mÐM +mÐN =180o By substitution: 90o+90o +mÐN =180o Solving leaves: mÐN =0o
- 13. Write an indirectproof: LMN LMN D D Given: Prove: has at most one right angle. mÐN =0o If: , This means that there is no triangle LMN. Which contradicts the given statement. ÐL and ÐM So the assumption that are both right angles must be false. Therefore DLMN has at most one right angle.
- 14. Lesson Quiz: PartI Solve each equation. Write a justification for each step. 1.
- 15. Lesson Quiz: PartII Solve each equation. Write a justification for each step. 2. 6r – 3 = –2(r + 1)
- 16. Lesson Quiz: PartIII Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. ÐDEF @ ÐDEF 5. AB @ CD, so CD @ AB.
- 17. Lesson Quiz: PartI Solve each equation. Write a justification for each step. 1. Given z – 5 = –12 Mult. Prop. of = z = –7 Add. Prop. of =
- 18. Lesson Quiz: PartII Solve each equation. Write a justification for each step. 2. 6r – 3 = –2(r + 1) Given 6r – 3 = –2r – 2 8r – 3 = –2 Distrib. Prop. Add. Prop. of = 6r – 3 = –2(r + 1) 8r = 1 Add. Prop. of = Div. Prop. of =
- 19. Lesson Quiz: PartIII Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. ÐDEF @ ÐDEF 5. AB @ CD, so CD @ AB. Trans. Prop. of = Reflex. Prop. of @ Sym. Prop. of @
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