1. Logic

2. Logical progression of thought A path others can follow and agree with Begins with a foundation of accepted In Euclidean Geometry begin with point, line and plane

4. Short sweet and to the point

5. Number Pattern Is this proof of how numbers were developed?

6. 2 = 1 a = b a 2 = ab a 2 - b 2 = ab-b 2 (a-b)(a+b) = b(a-b) a+b = b b+b = b 2b = b 2 = 1 Mathematical Proof

7. Geometry Undefined terms Are not defined, but instead explained. Form the foundation for all definitions in geometry. Postulates A statement that is accepted as true without proof. Theorem A statement in geometry that has been proved.

8. A form of reasoning that draws a conclusion based on the observation of patterns. Steps Inductive Reasoning Find counterexample to disprove conjecture 1.Identify a pattern 2.Make a conjecture

9. Does not definitely prove a statement, rather assumes it Educated Guess at what might be true Example Polling 30% of those polled agree therefore 30% of general population Inductive Reasoning

11. Not Proof

12. Find the next item in the pattern. Identifying a Pattern 7, 14, 21, 28, … The next multiple is 35. Multiples of 7 make up the pattern.

13. Find the next item in the pattern. Identifying a Pattern 4, 9, 16, … The next number is 25. Sums of odd numbers make up the pattern. 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 1 2 = 1 2 2 = 4 3 2 = 9 4 2 = 16 5 2 = 25

14. Find the next item in the pattern. Identifying a Pattern In this pattern, the figure rotates 90° counter-clockwise each time. The next figure is.

15. Complete the conjecture. Making a Conjecture The sum of two odd numbers is ?. The sum of two positive numbers is positive. List some examples and look for a pattern. 1 + 1 = 23.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917

16. Find the next item in the pattern. Identifying a Pattern January, March, May,... The next month is July. Alternating months of the year make up the pattern. The next month is… then August Perhaps the pattern was… Months with 31 days.

17. The product of two odd numbers is ?. Complete the conjecture. The product of two odd numbers is odd. List some examples and look for a pattern. 1  1 = 1 3  3 = 9 5  7 = 35

18. Counterexample - An example which disproves a conclusion Observation 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 are odd Conclusion All prime numbers are odd. 2 is a counterexample Inductive Reasoning

19. Show that the conjecture is false by finding a counterexample. Finding a Counterexample For every integer n, n 3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n 3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n 3 = –27 and –27  0, the conjecture is false. n = –3 is a counterexample.

20. Example 1 90% of humans are right-handed. Joe is a human. Example 2 Every life form that everyone knows of depends on liquid water to exist. Example 3 All of the swans that all living beings have ever seen are white. Inductive Reasoning Therefore, the probability that Joe is right-handed is 90%. Inductive reasoning allows for the possibility that the conclusion is false, even where all of the premises are true Therefore, all swans are white. Therefore, all known life depends on liquid water to exist.

21. Conjectures about our class….

22. Supplementary angles are adjacent. True? The supplementary angles are not adjacent, so the conjecture is false. 23 ° 157 °

23. Homework 2.1 and 2.2

24. To determine truth in geometry… Information is put into a conditional statement. The truth can then be tested. A conditional statement in math is a statement in the if-then form. If hypothesis, then conclusion A bi-conditional statement is of the form If and only if. If and only if hypothesis, then conclusion.

25. Underline the hypothesis twice The conclusion once 1.A figure is a parallelogram if it is a rectangle. 2.Four angles are formed if two lines intersect.

26. Determine if the conditional is true. If false, give a counterexample. Analyzing the Truth Value of a Conditional Statement You can have acute angles with measures of 80° and 30°. In this case, the hypothesis is true, but the conclusion is false. If two angles are acute, then they are congruent. Since you can find a counterexample, the conditional is false.

27. Analyzing the Truth Value of a Conditional Statement Determine if the conditional is true. “If a number is odd, then it is divisible by 3” If false, give a counterexample. An example of an odd number is 7. It is not divisible by 3. In this case, the hypothesis is true, but the conclusion is false. Since you can find a counterexample, the conditional is false.

28. For Problems 1 and 2: Identify the hypothesis and conclusion of each conditional. 1. A triangle with one right angle is a right triangle. 2. All even numbers are divisible by 2. 3. Determine if the statement “If n 2 = 144, then n = 12” is true. If false, give a counterexample. H: A number is even. C: The number is divisible by 2. H: A triangle has one right angle. C: The triangle is a right triangle. False; n = –12.

29. Identify the hypothesis and conclusion of each conditional. 1. A mapping that is a reflection is a type of transformation. 2. The quotient of two negative numbers is positive. 3. Determine if the conditional “If x is a number then |x| > 0” is true. If false, give a counterexample. H: A mapping is a reflection. C: The mapping is a transformation. H: Two numbers are negative. C: The quotient is positive. False; x = 0.

30. Different Forms of Conditional Statements Inverse: If an animal is not a cat, then it does not have 4 paws. Converse: If an animal has 4 paws, then it is a cat. Contrapositive: If an animal does not have 4 paws, then it is not a cat; True. Given Conditional Statement If an animal is a cat, then it has four paws. There are other animals that have 4 paws that are not cats, so the converse is false. There are animals that are not cats that have 4 paws, so the inverse is false. Cats have 4 paws, so the contrapositive is true.

31. A bi-conditional statement is of the form If and only if. If and only if hypothesis, then conclusion. Example A triangle is isosceles if and only if the triangle has two congruent sides.

32. Write as a biconditional Parallel lines are two coplanar lines that never intersect Two lines are parallel if and only if they are coplanar and never intersect.

33. Homework 2.3

34. To determine truth in geometry… Beyond a shadow of a doubt. Deductive Reasoning.

35. Deductive reasoning Uses logic to draw conclusions from Given facts Definitions Properties.

36. A pair of angles is a linear pair. The angles are supplementary angles. Two angles are complementary and congruent. The measure of each angle is 45. True or False And how do you know?

37. Modus Ponens Most common deductive logical argument p ⇒ q p ∴ q If p, then q p, therefore q Example If I stub my toe, then I will be in pain. I stub my toe. Therefore, I am in pain.

38. Modus Tollens Second form of deductive logic is p ⇒ q ~q ∴ ~p If p, then q not q, therefore not p Example If today is Thursday, then the cafeteria will be serving burritos. The cafeteria is not serving burritos, therefore today is not Thursday.

39. If-Then Transitive Property Third form of deductive logic A chains of logic where one thing implies another thing. p ⇒ q q ⇒ r ∴ p ⇒ r If p, then q If q, then r, therefore if p, then r Example If today is Thursday, then the cafeteria will be serving burritos. If the cafeteria will be serving burritos, then I will be happy. Therefore, if today is Thursday, then I will be happy.

40. Deductive reasoning Three forms p ⇒ qp ∴ q p ⇒ q~q ∴ ~p p ⇒ qq ⇒ r ∴ p ⇒ r

41. Draw a conclusion from the given information. If a polygon is a triangle, then it has three sides. If a polygon has three sides, then it is not a quadrilateral. Polygon P is a triangle. Conclusion: Polygon P is not a quadrilateral.

42. Homework 2.4

44. Proof 1.Algebraic 2.Geometric

45. Proof - argument that uses Logic Definitions Properties, and Previously proven statements to show that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid.

46. Algebraic Proof Properties of Real Numbers Equality Distributive Property a(b + c) = ab + ac. Substitution

47. Solve the equation 4m – 8 = –12. Write a justification for each step. Practice Solving an Equation with Algebra 4m – 8 = –12 Given equation +8 +8 Addition Property of Equality 4m = –4 Simplify. m =–1 Simplify. Division Property of Equality

48. Practice Solving an Equation with Algebra t = –14Simplify. Solve the equation. Write a justification for each step. Given equationMultiplication Property of Equality.

49. Solve for x. Write a justification for each step. Solving an Equation with Algebra NO = NM + MO 4x – 4 = 2x + (3x – 9) Substitution Property of Equality Segment Addition Post. 4x – 4 = 5x – 9 Simplify. –4 = x – 9 5 = x Addition Property of Equality Subtraction Property of Equality

50. Homework 2.5 Algebraic Proof

51. Geometric Proof Prove geometric theorems by using deductive reasoning. Two-column proofs.

52. Numbers are equal (=) and figures are congruent (). Remember!

53. When writing a proof: 1.Justify each logical step with a reason. 2.Each step must be clear enough so that anyone who reads your proof will understand them. Hypothesis Conclusion Definitions Postulates Properties Theorems

54. Proof Steps: 1.Start with given (hypothesis) 2.Logically connect given to conclusion Progressives statements with reasons 3.End with conclusion Two Column Proof – organizes your work StatementReason

55. Write a reason for each step, given that A and B are supplementary and mA = 45°. Writing Reasons Using a Two Column Proof 1. A and B are supplementary. mA = 45° Given 2. mA + mB = 180° Def. of supp s 3. 45° + mB = 180° Subst. Prop of = Steps 1, 2 4. mB = 135° Subtr. Prop of =

56. Writing Reasons Using a Two Column Proof Write a reason for each step, given that B is the midpoint of AC and AB  EF. 1. B is the midpoint of AC.Given 2. AB  BC 3. AB  EF 4. BC  EF Def. of mdpt. Given Trans. Prop. of 

57. Given: XY Prove: XY  XY Completing a Two-Column Proof StatementsReasons 1.1. Given 2. XY = XY2.. 3.. 3. Def. of  segs. Reflex. Prop. of = 

58. Given: 1 and 2 are supplementary, and 1  3 Prove: 3 and 2 are supplementary. Example 4 Completing a Two-Column Proof

59. Example 4 Continued StatementsReasons 1. 2.2.. 3..3. 4. 5. 1 and 2 are supplementary. 1  3 Given m1 + m2 = 180°Def. of supp. s m1 = m3 m3 + m2 = 180° 3 and 2 are supplementary Def. of  s Subst. Def. of supp. s

60. Given: 1, 2, 3, 4 Prove: m1 + m2 = m1 + m4 Use a Two Column Proof 1. 1 and 2 are supp. 1 and 4 are supp. 1. Linear Pair Thm. 2. m1 + m2 = 180°, m1 + m4 = 180° 2. Def. of supp. s 3. m1 + m2 = m1 + m4 3. Subst.

61. Homework 2.6 Geometric Proof

62. There are nine compositions (A to I) of eight colored cubes. Find two identical compositions. They can be rotated. There are nine compositions (A to I) of eight colored cubes. Find two identical compositions. They can be rotated.

63. Solution: Compositions D and I are identical.

64. There are nine compositions (A to I) of eight colored cubes. Find two identical compositions. They can be rotated. Four flat cubes Their patterns are drawn with bold black lines. Which can be drawn without taking your pencil off the paper or going along the same line twice? Which of them can't be drawn in this way?

65. There are nine compositions (A to I) of eight colored cubes. Find two identical compositions. They can be rotated. Shapes A and D can be drawn without taking your pencil off the paper or going along the same line twice. Shapes B and C can't be drawn in this way.