1. Chapter 2 Deductive 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. The If-Then Statement Conditional: is a two part statement with an actual or implied if-then. If p, then q. p ---> q hypothesis conclusion If I play football, then I am an athlete.

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

5. 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) All theorems, postulates, and definitions are conditional statements!!

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

7. Definition of Converse A conditional with the hypothesis and conclusion reversed. If q, then p. q ---> p hypothesis conclusion If I am an athlete, then I play football. Original: If the sun is shining, then it is daytime. **BE AWARE, THE CONVERSE IS NOT ALWAYS TRUE!!

8. Using the same hypothesis as the statement, but coming to a different conclusion. *Like a lawyer providing an alibi for his client… Definition of Counterexample

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

10. 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.

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

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

13. Write the converse of each statement If I play the tuba, then I am in the band. If I am in the band, then I play tuba. If 2x = 4, then x = 2 If x = 2, then 2x = 4

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

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

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

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

18. Provide a counterexample to show that each statement is false. If x 2 = 49, then x = 7.

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

20. WARM UP Is the original statement T or F? Then write the converse… if false, provide a counter example. If 3 points are in line, then they are colinnear. If 3 points are colinnear, then they are in line. If I live in Los Angeles, then I live in CA. If I live in CA, then I live in Los Angleles. –False, you could live in San Diego

21. 2.2 Properties from Algebra Objectives Do your first proof Use the properties of algebra and the properties of congruence in proofs

22. Properties from Algebra 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

23. Properties of Equality Addition Property Add prop of = Subtraction Property Subtr. Prop of = Multiplication Property Multp. Prop of = Division Property Div. Prop of = Substitution Property Substitution Numbers, variables, lengths, and angle measures WHAT I DO TO ONE SIDE OF THE EQUATION, I MUST DO …

24. Reflexive Property x = x. A number equals itself. Reflexive Prop. Transitive Property if x = y and y = z, then x = z. Two numbers equal to the same number are equal to each other. Transitive Pop. Properties of Equality Reflexive Property AB ≅ AB A segment (or angle) is congruent to itself Reflex. Prop Transitive Property If AB ≅ CD and CD ≅ EF, then AB ≅ EF Two segments (or angles) congruent to the same segment (or angle) are congruent to each other. Trans. Prop Properties of Congruence

25. Rules of Thumb…. Measurements are = –(prop. of equality) Figures are  –(Prop. of congruencey)

26. Whiteboards Page 40 –#’s 1 – 10

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

28. Day 2 - How to write a proof Walk-Thru of examples on page 38 and 39

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

30. Given : XZ = 20 YZ = 7 Prove: XY = 13 ZY X Your Second Proof ** Before we actually do this as a proof, lets make a verbal argument about why this is true.**

31. StatementsReasons 1. 2. 3. 4.

32. Given : L 2 = 50 Prove: L 1 is congruent L 3 2 1 3

33. StatementsReasons 1.1. given 2. 3. 4.

34. Given : WX = YZ Y is the midpoint of XZ Prove: WX = XY W ZY X ** Before we actually do this as a proof, lets make a verbal argument about why this is true.**

35. StatementsReasons 1. Y is the midpoint of XZ 1. Given 2. XY = YZ2. Def of midpoint 3. WX = YZ3. Given 4. WX = XY4. Substitution

36. Warm-up Page. 40 #12 Discuss with class

37. 2.3 Objectives Use the Midpoint Theorem and the Bisector Theorem Know the kinds of reasons that can be used in proofs

38. Being a lawyer… When making your case, you might reference laws, statutes, and/or previous cases in order to make your argument… YOU BETTER MAKE SURE YOU ARE REFERECING THE CORRECT ONES OR THE JUDGE WILL KICK YOU OUT OF THE COURTROOM!!

39. The Midpoint Theorem 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?

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

41. Whiteboards Pg. 45 # 1-9

42. Given: AB = CD Prove: AC = BD 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 STATEMENTS REASONS A B C D

43. QUIZ REVIEW Underline the hypothesis and conclusion in each statement Write a converse of each statement and tell whether it is true or false Provide a counter example to show that the statement is false Be able to complete a proof Name the reasons used in a proof (there are 5)

44. WARM – UP Answer true or false. If false, write a one sentence explanation. 1.The converse of a true statement is sometimes false. 2.Only one counterexample is needed to disprove a statement. 3.Properties of equality cannot be used in geometric proofs. 4.Postulates are deduced from theorems. 5.Every angle has only one bisector.

45. Draw diagram on bottom of page 51 to reference during lesson ( add a line to make vertical angles)

46. 2.4 Special Pairs of Angles Objectives Apply the definitions of complimentary and supplementary angles State and apply the theorem about vertical angles

47. Complimentary & Supplementary angles Rules that apply to either type.. 1.We are always referring to a pair of angles (2 angles).. No more no less 2.Angles DO NOT have to be adjacent 3.**Do not get confused with the angle addition postulate

48. Definition :Complimentary Angles If two angles add up to 90, then they are complimentary. If m  ABC + m  SXT = 90, then  ABC and  SXT are complimentary. S X T A B C  ABC is the complement of  SXT  SXT is the complement of  ABC

49. Definition: Supplementary Angles If two angles add up to 180, then the angles are supplementary. If m  ABC + m  SXT = 180, then  ABC and  SXT are supplementary. S X T A B C  ABC is the supplement of  SXT  SXT is the supplement of  ABC

50. Complimentary & Supplementary angles Rules that apply to either type.. 1.We are always referring to a pair of angles (2 angles).. No more no less 2.Angles DO NOT have to be adjacent 3.**Do not get confused with the angle addition postulate 4.In proofs, you must first prove two L ’s add up to 90 or 180 before saying they are comp or suppl. NEED TO BE EXPLICT!!

51. True or False m  A + m  B + m  C = 180, then  A,  B, and  C are supplementary.

52. A- Sometimes B – Always C - Never Two right angles are ____________ complementary.

53. Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. 1 2 3 4

54. Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. 1 2 3 4

55. Vertical Angles Two angles formed on the opposite sides of the intersection of two lines. 1 2 3 4 The only thing the definition does is identify what vertical angles are… NEVER USE THE DEFINITION IN A PROOF!!!

56. Theorem Vertical angles are congruent (The definition of Vert. angles does not tell us anything about congruency… this theorem proves that they are.) 1 2 3 4 **THIS THEOREM WILL BE USED IN YOUR PROOFS OVER AND OVER

57. White Board Practice Find the measure of a complement and a supplement of  T. m  T = 89

58. If  1 and  2 are vertical angles, m  1 = 2x+18 and m  2 = 3x+4, Find x. 14

59. White Board Practice A supplement of an angle is three times as large as a complement of the angle. Find the measure of the angle. Let x = the measure of the angle. 180 – x : This is the supplement 90 – x : This is the complement 180 – x = 3 (90 – x) 180 – x = 270 – 3x 2x = 90 x = 45

60. Whiteboard

61. Warm – Up Student will complete #33 from page 54 on front board

62. 2.5 Perpendicular Lines Objectives Recognize perpendicular lines Use the theorems about perpendicular lines

63. Perpendicular Lines (  ) If two intersecting lines form right angles, then they are perpendicular. If l  m, then the angles are right. l m If the angles are right, then l  m. What can you conclude about the rest of the angles in the diagram and why?

64. Two lines that form one right angle form four right angles The definition applies to intersecting rays and segments The definition can be used in two ways (bi- conditional) –PG. 56 Perpendicular Lines (  )

65. White Boards Page 57 –#1, 4, 5

66. White Boards Line AB  Line CD. G D F B C E A

67. 2.6 Planning a Proof Objectives Discover the steps used to plan a proof

68. Practice Given: m  1 = m  4 Prove: m  4 + m  2 = 180 12 4 3

69. ` StatementsReasons 1. m L 1 = m L 41. Given 2. 3. 4.

70. Practice Given: m  2 + m  3 = 180 Prove: m  1 = m  3 12 4 3

71. Given: x  m Prove: m L 1 + m L 2 = 90 x m 1 2 A B C