1. Logic

2. Logic 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. Logic
A crocodile steals a son from his father, and promises to return the child if the father can correctly guess what the crocodile will do.
What happens if the father guesses that the child will not be returned to him?
ANSWER: There is no solution.
If the crocodile keeps the child, he violates his rule, as the father predicted correctly.
If the crocodile returns the child, he still violates his rule as the father’s prediction was wrong.

7. Van Hiele Levels
theory that describes how students learn geometry
theory originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van Hiele (wife and husband) at Utrecht University in the Netherlands
Children at Level 0 will often say all of these shapes are triangles, except E, which is too "skinny".
They may say F is "upside down".
Students at Level 1 will recognize that only E and F are valid triangles

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

9. Mathematical
Proof
2 = 1
a = b a2 = ab a2 - b2 = ab-b (a-b)(a+b) = b(a-b) a+b = b b+b = b b = b = 1

10. 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.

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

12. Inductive Reasoning 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

14. Inductive Reasoning
Not Proof

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

16. 4, 9, 16, … Identifying a Pattern 12 = 1 22 = 4 1 = 1 32 = 9 1 + 3 = 4
Find the next item in the pattern.
4, 9, 16, …
Sums of odd numbers make up the pattern.
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
The next number is 25.
1 = 1
= 4
= 9
= 16
= 25

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

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

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

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

21. 2 is a counterexample Inductive Reasoning
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

22. Finding a Counterexample
Show that the conjecture is false by finding a counterexample.
For every integer n, n3 is positive.
Pick integers and substitute them into the expression to see if the conjecture holds.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the conjecture is false.
n = –3 is a counterexample.

23. even where all of the premises are true
Inductive Reasoning
What can you conclude?
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.
Therefore, the probability that Joe is right-handed is 90%.
Therefore, all known life depends on liquid water to exist.
Therefore, all swans are white.
Inductive reasoning allows for the possibility that the conclusion is false,
even where all of the premises are true

24. Conjectures about our class….

25. Logic
Common Sense

26. True? 157⁰ 23⁰ Supplementary angles are adjacent.
Based on the definition, supplementary angles sum to 180⁰. Therefore, supplementary angles can be non-adjacent, so the conjecture is false.

27. So that it looks like this:
Look at the triangle of six pennies below: I want to turn this triangle upside-down:
So that it looks like this:
What is the smallest number of coins I must move?
Two coins

28. A ship has a rope ladder hanging over its side.
The rungs are 230 mm apart (23 cm).
How many rungs will be underwater when the tide has risen one metre?
None of the rungs go underwater. WHY?

29. What mathematical symbol can be put between 5 and 9, to get a number bigger than 5 and smaller than 9?
A Decimal Point 5.9

30. Homework 2.1 and 2.2

31. If…

32. 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.

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

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

35. 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.

36. 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 n2 = 144, then
n = 12” is true. If false, give a counterexample.
H: A triangle has one right angle.
C: The triangle is a right triangle.
H: A number is even.
C: The number is divisible by 2.
False; n = –12.

37. 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.

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

39. 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.

40. 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.

41. Conditional Statements
Transition from hypothesis to p
from conclusion to q
Facilitates analysis of both parts.

42. Statement Pythagoras is Greek. A triangle has three sides.
Paris is the capital of Spain.

43. Writing Conditional Statements
Given p: You give me twenty dollars.
q: I will be your best friend.
Write the following statement in logic notation
"If you give me $20, then I will be your best friend"
p ->q q -> p
What of:
If you are my friend, I will give you $20.

44. Truth Values of Conditional Statements
If today is Friday, then tomorrow is Saturday.
If the month is October, then next month is November.
If you are 14 years old, then at your next birthday you will be 15 years old.
If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive.
If you
If you are in Mrs. Kapler’s Math class, you have homework for tonight.
If you live in a pineapple under the sea, then you are Sponge Bob.
If you are Sponge Bob, then you live in a pineapple under the sea.
If you are cold, then you will put on a sweatshirt.
If you did not eat breakfast, then you are hungry.
If you ate breakfast, then you are not hungry.
If you attend Washington Technology, then you are in 8th grade.
If you are in 8th grade, then you attend Washington Technology.
p – proposition
q – next letter
p q p->q
T T T
T F F
F T T
F F T

45. Truth Values of Conditional Statements
If today is Friday, then tomorrow is Saturday.
If the month is October, then next month is November.
If you are 14 years old, then at your next birthday you will be 15 years old.
If you are 13 years old, then at your next birthday you are eligible to receive a permit to drive.
If you play basketball, then you are as tall as Mr. Lott.
If you are in Mrs. Kapler’s Math class, you have homework for tonight.
If you live in a pineapple under the sea, then you are Sponge Bob.
If you are Sponge Bob, then you live in a pineapple under the sea.
If you are cold, then you will put on a sweatshirt.
If you did not eat breakfast, then you are hungry.
If you ate breakfast, then you are not hungry.
If you attend Washington Technology, then you are in 8th grade.
If you are in 8th grade, then you attend Washington Technology.
If you are in Geometry, then you are in 8th grade.

46. Truth Values of Conditional Statements
p q p->q
T T T
T F F
F T T
F F T
Given p: x is prime
q: x is odd
What is the truth value of a -> q when x = 2?
2. Given p: x is prime
What is the truth value of p -> q when x = 9?
False
True

47. Conditional (if, then) statements
If you give a mouse a cookie, then he will need a glass of milk.
If you give him a glass of milk, then he will ask for a straw.
I have no straws. Should I give the mouse a cookie?

48. Conditional Statements handout
If we have class outside today, we will need to wear jackets and sweatshirts.
If you have English next hour, then you will go to the English classroom.
If you don’t eat breakfast in the morning, you will be hungry in your morning classes.
If your teacher is an alien, then he has green skin.
If Mrs. Kapler retired last year, then she continued teaching.
If today is Friday, then tomorrow is Wednesday.

49. Converse, Inverse, Contrapositive
Conditional statement: p -> q
Converse: q -> p
Inverse: ~p -> ~q
Contrapositive: ~q -> ~p
Which statements are logically equivalent?

50. Homework 2.3

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

52. Deductive reasoning Uses logic to draw conclusions from Given facts
Definitions
Properties
Postulates
Theorems

53. True or False
And how do you know?
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 .

54. 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.

55. Modus Tollens
Second form of deductive logic is
p ⇒ q
~q ∴ ~p
If p, then q
not q (q is false), 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.

56. 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.

57. Deductive reasoning
Conditional p q
Converse q p
Biconditional p q
Law of Syllagism q ⇒ r ∴ p ⇒ r

58. Biconditional Biconditionals are often used to form definitions.
"[some fact] if and only if [another fact]“ is true
When the truth values of both facts are exactly the same -- BOTH TRUE or BOTH FALSE then biconditional is true.
Biconditional
Biconditionals are often used to form definitions.

59. Biconditional Definition
A triangle is isosceles if and only if the triangle has two congruent (equal) sides.
Analysis
If a triangle is isosceles, then the triangle has two congruent sides. (true)
If a triangle has two congruent sides, then the triangle is isosceles. (true)

60. Three lines are coplanar if and only if they lie in the same plane.
Analysis of Conditional and Converse
If three lines are coplanar, then they lie in the same plane.
If three lines lie in the same plane, then they are coplanar.

61. If and only if - iff
A triangle is equilateral iff each of its angles measures 60°.
Analysis of Conditional and Converse
If a triangle is equilateral then each of its angles measure 60°.
If all the angles of a triangle measure 60° then the triangle is equilateral.

62. x = 3 if and only if x2 = 9 Analysis
Conditional statement: If x = 3, then x2 = 9. True
Converse: If x2 = 9, then x = 3. False
The biconditional statement is false.

63. Can a Biconditional be Formed? If not, find a counterexample
If two points lie in a plane, then the line containing them lies in the plane.
Each of the following statements is true. Write the converse of each statement
and decide whether the converse is true or false. If the converse is true, combine
it with the original statement to form a true biconditional statement. If the
converse is false, state a counterexample.

64. Can a true Biconditional be Formed? If not, find a counterexample
Conditional: If two points lie in a plane, then the line containing them lies in the plane.
Converse: If a line containing two points lies in a plane, then the points lie in the plane. True – as shown in the diagram. So, it can be combined with the original statement to form the true biconditional statement
Biconditional statement: Two points lie in a plane if and only if the line containing them lies in the plane.
Each of the following statements is true. Write the converse of each statement
and decide whether the converse is true or false. If the converse is true, combine
it with the original statement to form a true biconditional statement. If the
converse is false, state a counterexample.

65. Can a true Biconditional be Formed? If not, find a counterexample
Conditional: If a number ends in 0, then the number is divisible by 5.
Converse: If a number is divisible by 5, then the number ends in False – counterexample x = So, it cannot be to formed into a true biconditional statement
Each of the following statements is true. Write the converse of each statement
and decide whether the converse is true or false. If the converse is true, combine
it with the original statement to form a true biconditional statement. If the
converse is false, state a counterexample.

66. Biconditional Truth Table
A biconditional is read as "[some fact] if and only if [another fact]" and is true when the truth values of both facts are exactly the same -- BOTH TRUE or BOTH FALSE.

67. Segment Addition Postulate is a true biconditional
If B lies between points A and C,
then AB + BC = AC.
Converse
If AB + BC = AC, then B lies between A and C.
Combining these statements produces the following true biconditional
Point B lies between points A and C
if and only if AB + BC = AC.

68. 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.

69. Homework 2.4 and 2.5

70. Proof
Algebraic
Geometric

71. Proof – logic-based argument that uses
Definitions
Properties
Postulates – accepted as true
Theorems - previously proven statements
to show that a conclusion is true.
Proof gives justification for each step.

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

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

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

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

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

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

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

79. Proof Steps:
Start with given (hypothesis)
Step by step logically connect given to conclusion. Justify each step with a reason. Why did you make this step?
End with conclusion
Two Column Proof – organizes your work
Statement Reason

80. Writing Reasons Using a Two Column Proof
Write a reason for each step, given that A and B are supplementary and mA = 45°.
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 =

81. 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
Def. of mdpt.
Given
3. AB  EF
4. BC  EF
Trans. Prop. of 

82. Completing a Two-Column Proof
Given: XY
Prove: XY  XY
Statements
Reasons 1.
1. Given
2. XY = XY
2. .
3. .
3. Def. of  segs.
Reflex. Prop. of = 

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

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

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

87. There are nine compositions (A to I) of eight colored cubes.
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.

88. Solution:
Compositions
D and I
are identical.

89. There are nine compositions (A to I) of eight colored cubes
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?

90. Shapes B and C can't be drawn in this way.
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.

91. Missing Square?