1. 2-1 Inductive Reasoning & Conjecture
INDUCTIVE REASONING is reasoning that uses a number of specific examples to arrive at a conclusion When you assume an observed pattern will continue, you are using INDUCTIVE REASONING. 1

2. 2-1 Inductive Reasoning & Conjecture
A CONCLUSION reached using INDUCTIVE REASONING is called a CONJECTURE. 2

3. 2-1 Inductive Reasoning & Conjecture
Example 1 Write a conjecture that describes the pattern in each sequence. Use your conjecture to find the next term in the sequence. 3

4. 2-1 Inductive Reasoning & Conjecture
Example 1a
What is the next term?
3, 6, 12, 24,
Conjecture:
Multiply each term by 2 to get the next term.
The next term is 24 •2 =
48. 4

5. 2-1 Inductive Reasoning & Conjecture
Example 1b What is the next term? 2, 4, 12, 48, 240 Conjecture: To get a new term, multiply the previous number by the position of the new number. The next term is
240•6 =
1440. 5

6. 2-1 Inductive Reasoning & Conjecture
Example 1c Conjecture: The number of small triangles is the perfect squares.
What is the next shape? ? 1 4 9 6

7. 2-1 Inductive Reasoning & Conjecture
The next big triangle should have _____ little triangles. 9 4 1 7

8. 2-1 Inductive Reasoning & Conjecture
EX 2 Make a conjecture about each value or geometric relationship. List or draw some examples that support your conjecture. 8

9. 2-1 Inductive Reasoning & Conjecture
EX 2a The sum of an odd number and an even number is __________. Conjecture: The sum of an odd number and an even number is ______________.
an odd number
Example: = 5
Example: = 73 9

10. 2-1 Inductive Reasoning & Conjecture
EX 2b For points L, M, & N, LM = 20, MN = 6, AND LN = 14. 20 L N M 14 6 10

11. 2-1 Inductive Reasoning & ConjectureL N M
Conjecture: N is between L and M. OR L, M, and N are collinear. 11

12. 2-1 Inductive Reasoning & Conjecture
Counterexample: An example that proves a statement false 12

13. 2-1 Inductive Reasoning & Conjecture
Give a counterexample to prove the statement false. If n is an integer, then 2n > n. One possible counterexample: n = – 8 because… 2(– 8) = – 16 and – 16 > – 8 is false!!! 13

14. 2-1 Inductive Reasoning & Conjecture
Assignment: p.93 – 96 (#14 – 30 evens, 40 – 44, 64 – 66 all) 14

15. 2-3 Conditional Statements
CONDITIONAL STATEMENT A statement that can be written in if-then form An example of a conditional statement: IF Portage wins the game tonight, THEN we’ll be sectional champs. 15

16. 2-3 Conditional Statements
HYPOTHESIS the part of a conditional statement immediately following the word IF CONCLUSION the part of a conditional statement immediately following the word THEN 16

17. 2-3 Conditional Statements
Example 1 Identify the hypothesis and conclusion of the conditional statement. a.) If a polygon has 6 sides, then it is a hexagon. Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon 17

18. 2-3 Conditional Statements
Example 1 (continued) b.) Joe will advance to the next round if he completes the maze in his computer game. Hypothesis: Joe completes the maze in his computer game Conclusion: he will advance to the next round 18

19. 2-3 Conditional Statements
Example 2 Write the statement in if-then form, Then identify the hypothesis and conclusion of each conditional statement. 19

20. 2-3 Conditional Statements
a.) A dog is Mrs. Lochmondy’s favorite animal. If-then form: If it is a dog, then it is Mrs. Lochmondy’s favorite animal. Hypothesis: it is a dog Conclusion: it is Mrs. Lochmondy’s favorite animal 20

21. 2-3 Conditional Statements
b.) A 5-sided polygon is a pentagon. If-then form: If it is a 5-sided polygon then it is a pentagon. Hypothesis: it is a 5-sided polygon Conclusion: it is a pentagon 21

22. 2-3 Conditional Statements
CONVERSE: The statement formed by exchanging the hypothesis and conclusion 22

23. 2-3 Conditional Statements
Example 3 Write the conditional and converse of the statement. Bats are mammals that can fly. Conditional: If it is bat, then it is a mammal that can fly. Converse: If it is a mammal that can fly, then it is a bat. 23

24. 2-3 Conditional Statements
Assignment: p (#18 – 30, evens 50, 52) For #50 & 52, only write the conditional & converse. 24

25. 2-4 Deductive Reasoning
Recall from section 2-1….. When you assume an observed pattern will continue, you are using INDUCTIVE REASONING. 25

26. 2-4 Deductive Reasoning
Deductive reasoning uses facts, rules, definitions, or properties to reach logical conclusions from given statements. 26

27. 2-4 Deductive Reasoning
EX 1 Determine whether each conclusion is based on inductive reasoning (a pattern) or deductive reasoning (facts). 27

28. 2-4 Deductive Reasoning
a.) In a small town where Tom lives, the month of April has had more rain than any other month for the past 4 years. Tom thinks that April will have the most rain this year. Answer: inductive reasoning – Tom is basing his reasoning on the pattern of the past 4 years. 28

29. 2-4 Deductive Reasoning
b.) Sondra learned from a metoerologist that if it is cloudy at night, it will not be as cold in the morning as if there were no clouds at night. Sondra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. Answer: deductive reasoning ‒ Sondra is using facts that she has learned about clouds and temperature. 29

30. 2-4 Deductive Reasoning
c.) Susan works in the attendance office and knows that all of the students in a certain geometry class are male. Terry Smith is in that geometry class, so Susan knows that Terry Smith is male. Answer: deductive reasoning – Susan’s reasoning is based on a fact she acquired while working in the attendance office. 30

31. 2-4 Deductive Reasoning
d.) After seeing many people outside walking their dogs, Joe observed that every poodle was being walked by an elderly person. Joe reasoned that poodles are owned exclusively by elderly people.
Answer: inductive reasoning –
Joe made his conclusion based on a pattern he saw.

32. 2-5 Postulates and Paragraph Proofs
Postulate or axiom –
a statement that is accepted as true without proof

33. 2-5 Postulates and Paragraph Proofs
Theorem ‒
A statement in geometry that has been proven

34. 2-5 Postulates and Paragraph Proofs
a logical argument in which each statement you make is supported by a statement accepted as true

35. 2-5 Postulates and Paragraph Proofs
Examples of Postulates
KNOW THESE POSTULATES!!!
2.1 Through any 2 points there is exactly one line.
2.2 Through any 3 noncollinear points there is exactly one plane.

36. 2-5 Postulates and Paragraph Proofs
2.3 A line contains at least 2 points.
2.4 A plane contains at least 3 noncollinear points.
2.5 If 2 points lie in a plane, then the entire line containing those 2 points lies in that plane.

37. 2-5 Postulates and Paragraph Proofs
2.6 If 2 lines intersect, then they intersect in exactly one point.
2.7 If 2 planes intersect, then they intersect in a line.

38. 2-5 Postulates and Paragraph Proofs
Example 2
Use the diagram on page 126 for these next examples.
Explain how the picture illustrates that each statement is true. Then state the postulate that can be used to show each statement is true.

39. 2-5 Postulates and Paragraph Proofs
a.) Points F and G lie in plane Q and on line m. Therefore, line m lies entirely in plane Q. Answer: Points F and G lie on line m, and the line lies in plane Q. (2.5) If 2 points lie in a plane, then the line containing the 2 points lies in the plane.

40. 2-5 Postulates and Paragraph Proofs
b.) Points A and C determine a line. Answer: Points A and C lie along an edge, the line that they determine. (2.1) Through any 2 points there is exactly one line.

41. 2-5 Postulates and Paragraph Proofs
c.) Lines s and t intersect at point D. Answer: Lines s and t of this lattice intersect at only one location point D. (2.6) If 2 lines intersect, then their intersection is exactly one point.

42. 2-5 Postulates and Paragraph Proofs
d.) Line m contains points F and G. Point E can also be on line m. Answer: The edge of the building is a straight line m. Points E, F, and G lie along this edge, so they line along line m. (2.3) A line contains at least 2 points.

43. 2-5 Postulates and Paragraph Proofs
Midpoint Theorem: If M is the midpoint of then .

44. 2-5 Postulates and Paragraph Proofs
Assignment: p.120(#10-15 all) p (#16-28 even, even)

45. 2-6 Algebraic Proof Day 1 See p
2-6 Algebraic Proof Day 1 See p.134 for the book’s way of explaining this.
PROPERTY
DESCRIPTION OF PROPERTY
Addition Property of Equality
Subtraction Property of Equality
Add the same value to
both sides of an equation.
Subtract the same value
from both sides of an equation.

46. 2-6 Algebraic Proof Day 1 DESCRIPTION OF PROPERTY PROPERTY
Multiplication Property of Equality
Division Property of Equality
Multiply both sides of an equation by the same nonzero value.
Divide both sides of an equation by the same nonzero value.

47. 2-6 Algebraic Proof Day 1 a = a If a =b, then b = a.
PROPERTY
DESCRIPTION OF PROPERTY
Reflexive Property of Equality
Symmetric Property of Equality
a = a
If a =b,
then b = a.

48. 2-6 Algebraic Proof Day 1 DESCRIPTION OF PROPERTY PROPERTY
Transitive Property of Equality
Substitution Property
If a = b and b = c, then a = c.
If a = b, then a can be substituted for b in any equation or expression.

49. DESCRIPTION OF PROPERTY Distributive Property
2-6 Algebraic Proof Day 1
PROPERTY
DESCRIPTION OF PROPERTY
Distributive Property
a(b + c) = ab + ac

50. 2-6 Algebraic Proof Day 1
EXAMPLES State the property that justifies each statement.

51. Division Property of Equality Multiplication Property of Equality
2-6 Algebraic Proof Day 1
EX 1
If 3x = 15, then x = 5.
Answer:
Division Property of Equality OR
Multiplication Property of Equality

52. Subtraction Property of Equality Addition Property of Equality
2-6 Algebraic Proof Day 1
EX 2 If 4x + 2 = 22, then 4x = 20.
Answer:
Subtraction Property of Equality OR
Addition Property of Equality

53. Transitive Property of Equality
2-6 Algebraic Proof Day 1
EX 3 If 7a = 5y and 5y = 3p, then 7a = 3p.
Answer:
Transitive Property of Equality

54. Transitive Property of Equality
2-6 Algebraic Proof Day 1
EX 4 If and , then
Answer:
Transitive Property of Equality

55. Reflexive Property of Equality
2-6 Algebraic Proof Day 1
EX 5
Answer:
Reflexive Property of Equality

56. Symmetric Property of Equality
2-6 Algebraic Proof Day 1
EX 6 If 3 = AB, then AB = 3.
Answer:
Symmetric Property of Equality

57. Substitution Property of Equality
2-6 Algebraic Proof Day 1
EX 7 If AB = 7.5 and CD = 7.5, then AB = CD.
Answer:
Substitution Property of Equality

58. 2-6 Algebraic Proof Day 1
Assignment:
p (#1-4,9-16,46-48,56-58)

59. 2-6 Algebraic Proof Day 2
EX 8 Write a justification for each step. Prove: If 2(5 – 3a) – 4(a + 7) = 92, then a = ‒11.

60. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2(5 – 3a) – 4(a + 7) = 92
Given
Distributive Prop

61. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS –10a – 18 = 92 +18 +18
Substitution Prop
Addition Prop

62. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS –10a = 110 a = –11
Substitution Prop
Division Prop
Substitution Prop

63. 2-6 Algebraic Proof Day 2
EX 9 Write a two-column proof to verify the conjecture. If 4(3x + 5) – 4x = 2(3x + 5) – 8, then x = –9.

64. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS
4(3x + 5) – 4x = 2(3x + 5) – 8
12x + 20 – 4x = 6x + 10 – 8
Given
Distributive
Prop

65. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS 8x + 20 = 6x + 2 –6x –6x
Substitution Prop
Subtraction Prop

66. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2x + 20 = 2 –20 –20
–20 –20
Substitution Prop
Subtraction Prop

67. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS 2x = –18 x = –9
Substitution Prop
Division Prop
Substitution Prop

68. 2-6 Algebraic Proof Day 2
EX 10 If the distance d an object travels is given by d = 20t + 5, the time t that the object travels is given by Write a two column proof to verify this conjecture.

69. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS d = 20t + 5 d – 5 = 20t
Given
Subtraction Prop

70. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS Division Property
Symmetric Prop

71. 2-6 Algebraic Proof Day 2
EX 11 Write a two-column proof to prove the conjecture. If , mB = 2mC, and mC = 45, then mA = 90.

72. 2-6 Algebraic Proof Day 2 mB = 2mC mC = 45 STATEMENTS REASONS
mA = mB
Given
Defn of ≅ s

73. 2-6 Algebraic Proof Day 2 STATEMENTS REASONS mA = 2mC mA = 2 (45)
Substitution Prop
Substitution Prop
Substitution Prop

74. 2-6 Algebraic Proof Day 2
Assignment: p.137 – 139(#5 – 7,17 – 19)

75. 2-6 Algebraic Proof Day 2

76. 2-6 Algebraic Proof Day 2

77. 2-6 Algebraic Proof Day 2

78. 2-6 Algebraic Proof Day 2