Chapter 2 Deductive Reasoning

1. Chapter 2Deductive Reasoning • Learn deductive logic • Do your first 2-column proof • New Theorems and Postulates **PUT YOUR LAWYER HAT ON!!

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. p ---> q hypothesis conclusion If the sun is shining, then it is daytime.

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

5. All theorems, postulates, and definitions are conditional statements!! Hidden If-Thens A conditional may not contain either if or then! Two intersecting lines are contained in exactly one plane. Which is the hypothesis? Which is the conclusion? two lines intersect exactly one plane contains them The whole thing: If two lines intersect, then exactly one plane contains them. (Theorem 1 – 3)

6. The Converse A conditional with the hypothesis and conclusion reversed. Original: If the sun is shining, then it is daytime. If q, then p. q ---> p hypothesis conclusion If it is daytime, then the sun is shining. **BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!

7. An if –then statement is false if an example can be found for which the hypothesis is true and the conclusion is false. The example is called the Counterexample. *Like a lawyer providing an alibi for his client… The Counterexample

8. The Counterexample If p, then q FALSE TRUE **You need only a single counterexample to prove a statement false.

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 **Definitions, Theorems and postulates have no counterexample. Otherwise they would not be true. To be true, it must always be true, with no exceptions.

10. Other Forms If p, then q p implies q p only if q q if p Conditional statements are not always written with the “if” clause first. All of these conditionals mean the same thing. What do you notice?

11. 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. The Biconditional Statement: If m1 = 90, then 1 is a right angle. Converse: If 1 is a right angle, thenm1 = 90. m1 = 90 if and only if 1 is a right angle. 1 is a right angle if and only if m1 = 90 .

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

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

14. Write the converse of each statement • If I play football, then I am an athlete • If I am an athlete, then I play football • If 2x = 4, then x = 2 • If x = 2, then 2x = 4

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

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

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

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

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

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 I’ll dive if you dive

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

23. Provide a counterexample to show that each statement is false. If x2 = 49, then x = 7.

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

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

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

27. 2.2 Properties from Algebra Objectives • Do your first proof • Use the properties of algebra and the properties of congruence in proofs

28. see properties on page 37 Read the first paragraph This lesson reviews the algebraic properties of equality that will be used to write proofs and solve problems. We treat the properties of Algebra like postulates Meaning we assume them to be true Properties from Algebra

29. Properties of Equality Numbers, variables, lengths, and angle measures WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …

31. Your First Proof Given: 3x + 7 - 8x = 22 Prove: x = - 3 (specifics)(general rules) 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. =

32. Properties of Congruence Segments, angles and polygons

33. Your Second Proof A B Given: AB = CD C D Prove: AC = BD STATEMENTS REASONS • 1. AB = CD 1. Given • 2. BC = BC 2. Reflexive prop. • 3. AB + BC = BC + CD 3. Addition Prop. = • 4. AB + BC = AC 4. Segment Addition Post. • BC + CD = BD • 5. AC = BD 5. Substitution

34. 2.3 Proving Theorems Objectives • Use the Midpoint Theorem and the Bisector Theorem • Know the kinds of reasons that can be used in proofs

35. YUMMY !

36. 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

37. 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.

38. 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.

39. 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.

40. 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.

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

42. If M is the midpoint of AB, then AM = ½ AB and MB = ½ AB How is the definition of a midpoint different from this theorem? One talks about congruent segments One talks about something being half of something else How do you know which one to use in a proof? The Midpoint Theorem

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

44. Given: M is the midpoint of AB Prove: AM = ½ AB and MB = ½ AB B M A Statements (specifics) Reasons (general rules) 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

45. 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

46. 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  XBC 2. Definition of Angle Bisector or  ABX  m  XBC 3.m ABX + m  XBC = 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 = 6. m  XBC = ½ m  ABC 6. Substitution Property

47. Given Information Definitions (bi-conditional) Postulates Properties of equality and congruence Theorems Reasons Used in Proofs (pg. 45)

48. How to write a proof(The magical steps) • Use these steps every time you have to do a proof in class, for homework, on a test, etc.

49. Example 1 Given : m  1 = m  2; AD bisects  CAB; BD bisects  CBA Prove: m  3 = m  4 C D 2 1 3 4 B A

50. 1. Copy down the problem. • Write down the given and prove statements and draw the picture. Do this every single time, I don’t care that it is the same picture, or that the picture is in the book. • Draw big pictures • Use straight lines