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Warm Up

Warm Up. Lesson Presentation. Lesson Quiz. 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.

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Warm Up

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  1. Warm Up Lesson Presentation Lesson Quiz

  2. 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-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 2are supplementary.

  4. Writing a Two-Column Proof : 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

  5. TEACH! Writing a Two-Column Proof 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.

  6. TEACH! Continued 1 and 2 are complementary. 2 and 3 are complementary. Given m1+ m2 = 90° m2+ m3 = 90° Def. of comp. s m1+ m2 = m2+ m3 Subst. Reflex. Prop. of = m2= m2 m1 = m3 Subtr. Prop. of = 1  3 Def. of  s

  7. Use indirect reasoning to 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 reasoning to 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. 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.

  9. Use indirect reasoning to 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 indirect proof 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 indirect proof: Indirect proof: Assume has more than one right angle. That is assume are both right angles.

  12. Write an indirect proof: If are both right angles, then According to the Triangle Angle Sum Theorem,. By substitution: Solving leaves:

  13. Write an indirect proof: If: , This means that there is no triangle LMN. Which contradicts the given statement. So the assumption that are both right angles must be false.

  14. Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1.

  15. Lesson Quiz: Part II Solve each equation. Write a justification for each step. 2.6r – 3 = –2(r + 1)

  16. Lesson Quiz: Part III Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. DEF  DEF 5. ABCD, so CDAB.

  17. Given z – 5 = –12 Mult. Prop. of = z = –7 Add. Prop. of = Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1.

  18. Given 6r – 3 = –2(r + 1) 6r – 3 = –2r – 2 Distrib. Prop. Add. Prop. of = 8r – 3 = –2 8r = 1 Add. Prop. of = Div. Prop. of = Lesson Quiz: Part II Solve each equation. Write a justification for each step. 2.6r – 3 = –2(r + 1)

  19. Lesson Quiz: Part III Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. DEF  DEF 5. ABCD, so CDAB. Trans. Prop. of = Reflex. Prop. of  Sym. Prop. of 

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