Chapter 2 Deductive Reasoning

1. Chapter 2Deductive Reasoning • Learn deductive logic • Do your first 2-column proof • New Theorems and Postulates

2. 2.1 If – Then Statements Objectives • Recognize the hypothesis and conclusion of an if-then statement • State the converse of an if-then statement • Use a counterexample • Understand if and only if

3. Conditional: is a two part statement with an actual or implied if-then. The If-Then Statement If p, then q. hypothesis conclusion If the sun is shining, then it is daytime.

4. Hidden If-Thens A conditional may not contain either if or then! All of my students love Geometry. If you are my student, then you love Geometry. Which is the hypothesis? Which is the conclusion? You are my student you love Geometry

5. The Converse A conditional with the hypothesis and conclusion reversed. Original: If the sun is shining, then it is daytime. If q, then p. hypothesis conclusion If it is daytime, then the sun is shining.

6. The Counterexample If p, then q FALSE TRUE

7. The only way a conditional can be false is if the hypothesis is true and the conclusion is false. This is called a counterexample. The Counterexample

8. The Counterexample • This is HARD !

9. The Counterexample If x > 5, then x = 6. If x = 5, then 4x = 20 x could be equal to 5.5 or 7 etc… always true, no counterexample

10. Group Practice • Provide a counterexample to show that each statement is false. If you live in California, then you live in La Crescenta.

11. Group Practice • Provide a counterexample to show that each statement is false. If AB  BC, then B is the midpoint of AC.

12. Group Practice • Provide a counterexample to show that each statement is false. If a line lies in a vertical plane, then the line is vertical

13. Group Practice • Provide a counterexample to show that each statement is false. If a number is divisible by 4, then it is divisible by 6.

14. Group Practice • Provide a counterexample to show that each statement is false. If x2 = 49, then x = 7.

15. Other Forms • If p, then q • p implies q • p only if q • q if p What do you notice?

16. White Board Practice • Circle the hypothesis and underline the conclusion VW = XY implies VW  XY

17. Circle the hypothesis and underline the conclusion VW = XY implies VW  XY

18. Circle the hypothesis and underline the conclusion K is the midpoint of JL only if JK = KL

19. Circle the hypothesis and underline the conclusion K is the midpoint of JL only if JK = KL

20. Circle the hypothesis and underline the conclusion n > 8 only if n is greater than 7

21. Circle the hypothesis and underline the conclusion n > 8 only if n is greater than 7

22. Circle the hypothesis and underline the conclusion I’ll dive if you dive

23. Circle the hypothesis and underline the conclusion I’ll dive if you dive

24. Circle the hypothesis and underline the conclusion If a = b, then a + c = b + c

25. Circle the hypothesis and underline the conclusion If a = b, then a + c = b + c

26. Circle the hypothesis and underline the conclusion If a + c = b + c, then a = b

27. Circle the hypothesis and underline the conclusion If a + c = b + c, then a = b

28. Circle the hypothesis and underline the conclusion r + n = s + n if r = s

29. Circle the hypothesis and underline the conclusion r + n = s + n if r = s

30. The Biconditional If a conditional and its converse are the same (both true) then it is a biconditional and can use the “if and only if” language. If m1 = 90, then 1 is a right angle. If1 is a right angle, then m1 = 90. m1 = 90 if and only if 1 is a right angle. m1 = 90 iff 1 is a right angle.

31. 2.2 Properties from Algebra Objectives • Do your first proof • Use the properties of algebra and the properties of congruence in proofs

32. see properties on page 37 Properties from Algebra

33. Properties of Equality

35. Properties of Congruence

36. Your First Proof Given: 3x + 7 - 8x = 22 Prove: x = - 3 STATEMENTS REASONS 1. 3x + 7 - 8x = 22 1. Given 2. -5x + 7 = 22 2. Substitution 3. -5x = 15 3. Subtraction Prop. = 4. x = - 3 4. Division Prop. =

37. Your Second Proof A B Given: AB = CD C D Prove: AC = BD STATEMENTS REASONS 1. AB = CD 1. Given 2. AB + BC = BC + CD 2. Addition Prop. = 3. AB + BC = AC 3. Segment Addition Post. BC + CD = BD 4. AC = BD 4. Substitution

38. 2.3 Proving Theorems Objectives • Use the Midpoint Theorem and the Bisector Theorem • Know the kinds of reasons that can be used in proofs

39. YUMMY !

40. PB & J Sandwich • How do I make one? • Pretend as if I have never made a PB & J sandwich. Not only have I never made one, I have never seen one or heard about a sandwich for that matter. • Write out detailed instructions in full sentences • I will collect this

41. First, open the bread package by untwisting the twist tie. Take out two slices of bread set one of these pieces aside. Set the other in front of you on a plate and remove the lid from the container with the peanut butter in it.

42. Take the knife, place it in the container of peanut butter, and with the knife, remove approximately a tablespoon of peanut butter. The amount is not terribly relevant, as long as it does not fall off the knife. Take the knife with the peanut butter on it and spread it on the slice of bread you have in front of you.

43. Repeat until the bread is reasonably covered on one side with peanut butter. At this point, you should wipe excess peanut butter on the inside rim of the peanut butter jar and set the knife on the counter.

44. Replace the lid on the peanut butter jar and set it aside. Take the jar of jelly and repeat the process for peanut butter. As soon as you have finished this, take the slice of bread that you set aside earlier and place it on the slice with the peanut butter and jelly on it, so that the peanut butter and jelly is reasonably well contained within.

45. If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB The Midpoint Theorem B A M

46. Important Notes • Does the order matter? • Don’t leave out steps  Don’t ASSume

47. Given: M is the midpoint of AB Prove: AM = ½ AB and MB = ½ AB B M A Statements Reasons 1. M is the midpoint of AB 1. Given 2. AM  MB or AM = MB 2. Definition of a midpoint 3. AM + MB = AB 3. Segment Addition Postulate 4. AM + AM = AB 4. Substitution Property Or 2 AM = AB 5. AM = ½ AB 5. Division Property of Equality 6. MB = ½ AB 6. Substitution

48. If BX is the bisector of ABC, then m  ABX = ½ m  ABC m  XBC = ½ m  ABC The Angle Bisector Theorem A X B C

49. A Given: BX is the bisector of ABC Prove: m  ABX = ½ m  ABC m  XBC = ½ m  ABC X B C 1. BX is the bisector of  ABC; 1. Given 2. m  ABX = m  BXC 2. Definition of Angle Bisector or  ABX  m  BXC 3.m ABX + m BXC = m ABC 3. Angle Addition Postulate 4.m ABX + m ABX = m ABC 4. Substitution or 2 m  ABX = m  ABC 5. m  ABX = ½ m  ABC 5. Division Property of Equality 6. m  BXC = ½ m  ABC 6. Substitution Property

50. Given Information Definitions Postulates (including Algebra) Theorems Reasons Used in Proofs