1. GEOMETRY 4-5 Using indirect reasoning Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. GEOMETRY 4-5 Using indirect reasoning TEACH! Identifying Algebraic Properties Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4b. 94° = 94° 4c. 0 = a, and a = x. So 0 = x. 4d. A  Y, so Y  A Sym. Prop. of = Reflex. Prop. of = Trans. Prop. of = Sym. Prop. of  Homework 1-16 Page 209 CH 4-5

3. GEOMETRY 4-5 Using indirect 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

4. GEOMETRY 4-5 Using indirect reasoning 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. Hypothesis Conclusion Definitions Postulates Properties Theorems

5. GEOMETRY 4-5 Using indirect reasoning

6. GEOMETRY 4-5 Using indirect reasoning

7. GEOMETRY 4-5 Using indirect reasoning 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.

8. GEOMETRY 4-5 Using indirect reasoning Fill in the blanks to complete the two-column proof. Given: XY Prove: XY  XY Example 2: Completing a Two-Column Proof StatementsReasons 1.1. Given 2. XY = XY2.. 3.. 3. Def. of  segs. Reflex. Prop. of = 

9. GEOMETRY 4-5 Using indirect reasoning If a diagram for a proof is not provided, draw your own and mark the given information on it. But do not mark the information in the Prove statement on it. Helpful Hint

10. GEOMETRY 4-5 Using indirect reasoning Use the given plan to write a two-column proof. Writing a Two-Column Proof from a Plan 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.

11. GEOMETRY 4-5 Using indirect reasoning Writing a Two-Column Proof : Continued StatementsReasons 1. 2.2.. 3..3. 4. 5. 1 and 2 are supplementary. 1  3 Given m1 + m2 = 180°Def. of supp. s m1 = m3 m3 + m2 = 180° 3 and 2 are supplementary Def. of  s Subst. Def. of supp. s

12. GEOMETRY 4-5 Using indirect reasoning Use the given plan to write a two-column proof if one case of Congruent Complements Theorem. TEACH! Writing a Two-Column Proof 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.

13. GEOMETRY 4-5 Using indirect reasoning TEACH! Continued StatementsReasons 1. 2.2.. 3..3. 4. 5. 6. 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 m2 = m2 m1 = m3 Subst. Reflex. Prop. of = Subtr. Prop. of = 1  3 Def. of  s

14. GEOMETRY 4-5 Using indirect reasoning 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.

15. GEOMETRY 4-5 Using indirect reasoning 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.

16. GEOMETRY 4-5 Using indirect reasoning 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. 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. Therefore, at least one of the items costs more than $25.

17. GEOMETRY 4-5 Using indirect reasoning 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

18. GEOMETRY 4-5 Using indirect reasoning Write an indirect proof: Indirect proof: Assume has more than one right angle. That is assume are both right angles.

19. GEOMETRY 4-5 Using indirect reasoning Write an indirect proof: If are both right angles, then According to the Triangle Angle Sum Theorem,. By substitution: Solving leaves:

20. GEOMETRY 4-5 Using indirect reasoning 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.

21. GEOMETRY 4-5 Using indirect reasoning

22. GEOMETRY 4-5 Using indirect reasoning Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1.

23. GEOMETRY 4-5 Using indirect reasoning Lesson Quiz: Part II Solve each equation. Write a justification for each step. 2. 6r – 3 = –2(r + 1)

24. GEOMETRY 4-5 Using indirect reasoning 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.

25. GEOMETRY 4-5 Using indirect reasoning Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1. z – 5 = –12Mult. Prop. of = z = –7Add. Prop. of = Given

26. GEOMETRY 4-5 Using indirect reasoning Lesson Quiz: Part II 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 = 1Add. Prop. of = Div. Prop. of =

27. GEOMETRY 4-5 Using indirect reasoning 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 