1. 2-1 Inductive Reasoning and Conjecturing

2. I. Study of Geometry Inductive reasoning Conjecture Counterexample

3. II. Form Given: Conjecture:

4. 1.Given: AB = BC Conjecture: 2.Given: <D and <F are supplementary And <E and <F are supplementary Conjecture: III. Problems

5. 3.Given collinear points D,E,F Conjecture: DE + EF = DF 4.For points A,B,C,D, AB=5, BC=10, CD=8, AD=12. Make a conjecture and draw a figure to illustrate.

6. 5. Make a conjecture about the next figure in the pattern.

7. 6. Find a counterexample for the following statement based on the graph. The mortgage rates for March decreased from February.

8. 7. For points A, B and C, AB = 6, BC = 10 and AC = 15. Make a conjecture and draw a figure to illustrate your conjecture.

9. Class work p. 63 1- 21 all

10. 2-2/ 2-3 If-Then Statements and Postulates

11. I. Conditional Statements If-then statements are used to clarify confusing statements Conditionals are if-then statements If If is the hypothesis Then Then is the conclusion Form: “If p, then q” is represented by p q “If q then p.” q p

12. Converse: switch the hypothesis and conclusion q p

13. Negation: not p ~ p

14. Inverse: a negation “If not p then not q” ~p ~q

15. Contrapositive: “If not q, then not p” ~q ~ p

16. Venn diagram:

17. Truth Value The truth value of a statement is whether it is true or false. Halle Berry is catwoman.

18. II. Problems 1.Write the statement “An angle of 40 is acute” in if-then form. 2. Identify the hypothesis and the conclusion: If it is Tuesday, then Phil plays tennis.

19. 3. Write the converse of:” An angle that measures 120 is obtuse.” Determine if the converse if true or false. If false, give a counterexample. 4. Write the inverse and contrapositive of the true conditional: “ If aliens have visited earth, then there is life on other planets.”

20. 5. Identify the hypothesis and conclusion. If you are an NBA basketball player, then you are at least 5’2’’ tall.

21. 7. Determine the truth value of the following statement for each set of conditions. If you pass the driver’s test, then you will get a driver’s license. You pass the driver’s test; you get a driver’s license You pass the driver’s test; you get a learner’s permit. You fail the driver’s test: you get a license.

22. 6. Write in conditional form. Adjacent angles have a common vertex. Perpendicular lines form right angles.

23. 7. Write the converse, inverse, and contrapositive of the statement All squares are quadrilaterals. Determine whether each statement is true or false. If a statement is false, give a counterexample

24. IV. Conclusion Five people arrive at meeting. Each one shakes hands one time with each of the others. How many different handshakes take place? Draw a diagram to illustrate.

25. Class work P. 78 1-15 all

26. 2-4 Deductive Reasoning I. Sherlock Holmes If Colonel Moran is the murderer, then he has powder burns on his shirt. The colonel does not have powder burns on his shirt. Therefore…

27. II. Logical reasoning Deductive reasoning is logical, if certain statements are true, and then other statements can be shown to follow from them

28. Law of detachment: If p q is a true conditional and p is true, then q is true.

29. Law of syllogism: If p q and q r are true conditional, then p r is also true.

30. III. Examples 1. Find the number of angles formed by 10 distinct lines with a common intersection point. 2. If a vehicle is a car, then it has four wheels is a true conditional. A sedan is a car. Use the law of detachment to reach a valid conclusion.

31. 3. Determine if a valid conclusion can be reached from the following true stamens: If I watch TV, I will not do my homework and I did my homework. Use the law of detachment. 4. If Elena takes the car to the store, she will stop at the post office and If Elena stops at the post office, she will buy stamps. Use the law of syllogism.

32. 5. The following is a true conditional. Determine whether each conclusion is valid based on the given information. Explain your reasoning. If a point is a midpoint of a segment, then it divides the segment into two congruent segments. Given:W is the midpoint of DC. Conclusion: DW  WC

33. 6. Use the Law of Syllogism to determine whether a valid conclusion can be reached from each set of statements. a.(1) If a figure is a square, then it has four right angles. (2) A figure with four right angles is a rectangle. b.(1) Perpendicular lines form right angles. (2) The sum of complementary angles equals 90.

34. 7. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. a.(1) If two angles form a linear pair, then they are supplementary. (2) If two angles are supplementary, then the sum of the angles equals 180. (3) If two angles form a linear pair, then the sum of the angles equals 180. b.(1) If angles are supplementary, the sum of their measures is 180. (2)  A and  B are supplementary. (3)  A is obtuse and  B is acute.

35. Class work P. 84 1- 11 all

36. 2-5 Postulates and Paragraph Proofs Postulate:

37. Postulates Through any two points, there is exactly one line. Through any three points not on the same line, there is exactly one plane. A line contains at least two points. A plane contains at least three points not on the same line. If two points lie in a plane, then the entire line containing those two points lies in that plane. If two planes intersect, then their intersection is a line.

38. Proof A logical argument in which each statement made is supported by a statement everyone knows is true!

39. Proofs There are 3 basic ways to proving logical arguments: Direct proofs conditional proofs indirect proofs

40. Paragraph/Informal Proof Write a paragraph to explain why a conjecture is true. A direct proof.

41. Five parts of a proof State the idea to be proved List the given information Draw a diagram or illustration State what it is to be proved Develop a system of inductive reasoning

42. Problems 1. Donna is setting up a network for her company. There are 7 computers in her office. She wants to connect each computer to every other computer so that if one computer fails, the others are still connected. How many connections does Donna have to make?

43. 2. Determine whether each statement is always, sometimes or never true. Explain. a.The intersection of plane M and plane N is point A. b.If A and B lie in plane W, then lies in plane W. c. lines in plane M.

44. 3. Write a paragraph proof. Given that is the angle bisector of  CAD, write a paragraph proof to show that  CAB   DAB.

45. Class work P. 91 1-15

46. 2-6 Algebraic Proof 2-7 Proving Segment Relationships Properties of equality Reflexive Symmetric Transitive Add/sub Mult/div Substitution distributive

47. Each statement is in the left-hand column. Each reason is in the right. TWO COLUMN PROOF

49. REASONS GIVEN POSTULATES THEOREMS PROPERTIES DEFINITIONS RULES

50. Congruence of segments is reflexive, symmetric, and transitive. Reflexive Symmetric Transitive II. Segment Congruence

51. Midpoint theorem

52. III. Problems 1. Name the property of equality that justifies each statement. If 3x=120, then x=40. If 12=AB, then AB=12. If AB=BC, and BC=CD, then AB=CD. If y=75 and y=m<A, then m<A=75.

53. 2. Justify each step in solving x/3 +4=1. StatementsReasons 1. x/3+4=1Given 2. x/3=-3 3. x=-9

54. 3. Verify Algebraic Relationships Solve 4(x + 3) = 52.

55. 4. Write a two-column proof. a.If 2(3x + 4) = 56, then x = 8 b. Given:2/3 + n = 9 -1/4 n Prove:n = 20/3

56. 5. If  A   B, and  B   C, then which of the following is a valid conclusion? Im  A = m  B and m  B = m  C II  A   C III  A is complementary to  C AII onlyBI and II CI, II and IIIDI and III

57. 6. On a clock, the angle formed by the hands at 4:00 is a 120° angle. If the angle formed at 4:00 is congruent to the angle formed at 8:00, prove that the angle formed at 8:00 is a 120° angle. Given:m  4 = 120  4   8 Prove:m  8 = 120

58. 8. Prove the following. Given:WX = YZ Prove:WY = XZ

59. 9. Prove the following. Given: AB  BC, AD  DE, BC  AD Prove:DE  AB

60. Class work P. 97 1-13 103 1- 11

61. 2-8 Verifying Angle Relationships I. Theorems If 2 angles form a linear pair, then they are supplementary. Congruence of angles is reflexive, symmetric, and transitive. Angles supplementary to the same angle or to congruent angles are congruent.

62. Angles complementary to the same angle or to congruent angles are congruent. All right angles are congruent. Vertical angles are congruent. Perpendicular lines intersect to form four right angles.

63. Problems 1. If m  XYZ = 122 and m  XYW = 86, find m  WYZ.

64. 2. If  1 and  2 form a linear pair and m  1 = 72, find m  2.

65. 3. In the figure,  1 and  2 form a linear pair and  1and  3 are supplementary. Prove that  2 and  3 are congruent.

66. 4. If  1 and  2 are vertical angles and m  1 = n and m  2 = 188 - 3n, find m  1 and m  2.

67. Class work P. 111 1-15