1. Chapter 2
Reasoning and Proof

2. Patterns and Inductive Reasoning
Lesson 2.1
Patterns and Inductive Reasoning

3. Introduction
Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. 21
blocks 15
blocks 10
blocks 6
blocks
... 3
blocks 1
block

4. Example
Describe a pattern in the sequence of numbers and predict the next number.
Pattern: Each number is 4 times the previous number
Pattern: Each number is 3 more than the previous number
a) 1, 4, 16, 64, ...
256
b) 2, 5, 8, 11, ... 14

5. 2 Types of Reasoning
Inductive Reasoning – Looking at several previous examples and drawing a conjecture (educated guess)
Deductive Reasoning – Uses facts, definitions, and other rules in order to draw a conclusion.

6. Using Inductive Reasoning
Look for a Pattern – Look at several examples. Pictures and tables can be helpful.
Make a Conjecture – Use examples to make a conjecture.
Verify the Conjecture – Use deductive reasoning to the conjecture is true for all cases.

7. Example
Complete the conjecture based on the pattern you observe in the specific cases.
1 + 1 = = = = 32
Conjecture: The sum of any two odd numbers is ________.

8. Counterexample An example that proves a statement false.
Ex. All four legged animals are dogs.
counterexample: cats
Ex. All red trucks are fire trucks.
counterexample: Dodge Ram

9. Homework
Page 85
6, 7,12, 13, 16, 20, 21-24, 25-28, 33-37, 38, 39

10. Conditional Statements
Lesson 2.2
Conditional Statements

11. Conditional statements
Conditional statement- a sentence that contains an “if” and a “then” phrase
Ex. If you frown, then you are sad
Ex. If an angle is right, then it is 90°

12. Hypothesis / Conclusion
Hypothesis-the phrase that follows the word if
Conclusion-the phrase that follows the word then
Ex. If you frown, then you are sad
Ex. If an angle is right, then it is 90°
Ex. You are late if you arrive after the bell

13. How to rewrite in if-then form
Identify the hypothesis and conclusion (look for an “if”).
Rewrite in the form: If hypothesis then conclusion.
Verify that the new statement has the same meaning as the initial statement.

14. Rewrite in if-then form
Three points are collinear if they lie on the same line
If they lie on the same line then three points are collinear
If three points lie on the same line then they are collinear
Cats make Ms. Wilson sneeze
If cats then make Ms. Wilson sneeze
If an animal is a cat then it makes Ms. Wilson sneeze

15. Different forms a conditional
Inverse – negate both hypothesis and conclusion
Converse – switch the places of the hypothesis and conclusion
Contrapositive – negate both the hypothesis and conclusion and then switch their places (basically do inverse then converse)

16. (implies) Symbolic Notation p  q means “p implies q”
p is the hypothesis and q is the conclusion
It can be rewritten as “If p then q”

17. Symbolic Notation
negation is a ~ given conditional: p  q inverse: ~ p  ~ q converse: q  p contrapostive: ~ q  ~ p

18. Negation Negation - change the “truth” Ex. The letter A is a vowel
Done by:
changing a word to its opposite
inserting (or removing) the word NOT
Ex. The letter A is a vowel
Negation: The letter A is NOT a vowel
The letter A is a consonant

19. True or False A conditional statement can either be true or false
To show true - must show true for ALL cases
To show false - must provide one time false called a counterexample

20. Writing the inverse, converse and contrapositive
Given conditional: If an animal is a fish then it can swim
First identify the hypothesis and conclusion
Inverse:
Converse:
Contrapositive:

21. Is the conditional true or false? If false, provide a counterexample
If two angles are a linear pair then they are supplementary angles
If a number is odd, then it is divisible by 3

22. Homework
2-2 Practice (on website)
Page 93
13-19

23. Biconditional Statements and Definitions
Lesson 2.3
Biconditional Statements and Definitions

24. Biconditional statement
A statement that contains “if and only if” (symbol )
The same as writing a statement and its converse all in one.
To be “true” - statement and its converse must both be true

25. Definitions as conditionals
Definitions can be written word first or description first
A right angle has a measure of 90°
If an angle is right then it has a measure of 90°
If an angle has a measure of 90° then it is right
All definitions can be rewritten as a true bi- conditional statement
An angle is right if and only if it has a measure of 90°

26. Practice writing statements
Perpendicular lines are two lines that intersect to form a right angle
Write as a conditional statement:
Write as its converse:
Write as a bi-conditional statement:

27. More Practice X = 5 if and only if 2x = 10
What kind of statement is this?
Re-write as a conditional.
Write the converse of this conditional.

28. Homework
Page 101
7-10, 13-15, 24, 25

29. Lesson 2-4
Deductive Reasoning

30. Laws of Logic Law of Ruling Out Possibilities
Given: A or B is true and B is not true
Our conclusion: A is true

31. Laws of Logic Law of Detachment Given: pq is true and p is true
Our conclusion: q is true

32. Laws of Logic Law of Syllogism (Transitivity) Given: pq and qr
Our conclusion: pr
look for common phrase in one hypothesis and one conclusion

33. Example If the sun is out, then it is not cloudy The sun is out
Our conclusion:
It is not cloudy

34. Example If trees are around, then Ms. Wilson sneezes
Ms. Wilson sneezed
Our conclusion:
NO conclusion
(We can only make a conclusion if we know that trees are around)

35. Example If the sun is out, then we’ll go biking
If we go biking, then we’ll be happy
Our conclusion:
If the sun is out, then we’ll be happy

36. Example If it is Friday, then there is a game tonight
If it is Friday, then tomorrow is Saturday
Conclusion:
NO conclusion
(The common phrase must be in one hypothesis and one conclusion)

37. Homework

38. Reasoning in Algebra and Geometry
Lesson 2-5
Reasoning in Algebra and Geometry

39. Properties from Algebra
Addition Property
if a = b, then a + c = b + c
Subtraction Property
if a = b, then a – c = b – c
Multiplication Property
if a = b, then a * c = b * c
Division Property
if a = b & c is not 0, then a / c = b / c

40. Properties from Algebra
Distributive Property
if a ( b + c ) then a * b + a * c
Substitution Property
if a = b, then a can be substituted for b in any equation or expression

41. Properties from Algebra
Reflexive Property
for any real number, a = a
Symmetric Property
if a = b, then b = a
Transitive Property
if a = b and b = c, then a = c

42. Example Multiplication Property
State the property being demonstrated with each statement
If x + 10 = 15 then x = 5
Subtraction Property
If AB = 4cm then 5AB = 20cm
Multiplication Property

43. Example Transitive Property MN
Use the property to complete the statement
Transitive Property
If RB = MN and ____ = WY then RB = WY MN

44. Example Given Addition Property Subtraction Property Division Property
Solve the equation and write a reason for each step
3x - 17 = x + 5
3x = x + 22
2x =
x =
Given
Addition Property
Subtraction Property
Division Property

45. Example Solve the equation and write a reason for each step
55x – 3(9x + 12) = -64
55x – 27x – = -64
28x = -64
28x = -28
x = -1
Given
Distributive Property
Combine Like Terms
Addition Property
Division Property

46. Homework
Page 117
6-20

47. Proving Angles Congruent
Lesson 2-6
Proving Angles Congruent

48. Proofs & Theorems No need to write this down
Proof - statements and reasons that show the logical order of an argument
Theorem – a statement that can be proven true

49. Steps of a Proof Draw the picture Make a “T” chart
Statements on left / Reasons on right
Input the given information
Mark the drawing
Make statements and reasons that lead to the “prove” statement

50. Special Types of Angles review from Chapter 1
Right – has a measure of 90°
Complementary - 2 angles whose sum is 90°
Supplementary - 2 angles whose sum is 180°

51. Vertical Angle Theorem
Linear Pair Postulate
If two angles form a linear pair, then they are supplementary
Vertical Angle Theorem
Vertical angles are congruent

52. Example Solve for x 2x + 18 = 6x - 190 Why? Vertical s are =

53. Complete the proof Statements Reason 2 1 3 Given
Given: m1 = m2
2 and 3 are supplementary
Prove: 1 and 3 are supplementary
Statements Reason
m1 = m2
m 2 + m 3 = 180°
m 1 + m 3 = 180°
1 and 3 are supplementary 2 1 3
Given
Def of supplementary s
Substitution Property

54. Statements Reasons 6 8 5 7 Given: 6  7 Prove: 5  8 6  7
5  6
5  7
7  8
5  8 5 7
Given
Vertical  Theorem
Transitive Property

55. Homework

56. Law of Detacment -What does this mean?
Given:
One if-then sentence
The “if” part is repeated
Conclusion:
The “then” part

57. Law of Syllogism - What does this mean?
Given:
Two if-then sentence
A common phrase is once in the “if” part and once in the “then” part
Conclusion:
The non-common parts are put together into one if-then sentence