1. Chapter 2
Reasoning and Proof

2. Chapter Objectives Recognize conditional statements
Compare bi-conditional statements and definitions
Utilize deductive reasoning
Apply certain properties of algebra to geometrical properties
Write postulates about the basic components of geometry
Derive Vertical Angles Theorem
Prove Linear Pair Postulate
Identify reflexive, symmetric and transitive

3. Conditional Statements
Lesson 2.1
Conditional Statements

4. Lesson 2.1 Objectives Analyze conditional statements
Write postulates about points, lines, and planes using conditional statements

5. Conditional Statements
A conditional statement is any statement that is written, or can be written, in the if-then form.
This is a logical statement that contains two parts
Hypothesis
Conclusion
If today is Tuesday, then tomorrow is Wednesday.

6. Hypothesis
The hypothesis of a conditional statement is the portion that has, or can be written, with the word if in front.
When asked to identify the hypothesis, you do not include the word if.
If today is Tuesday, then tomorrow is Wednesday.

7. Conclusion
The conclusion of a conditional statement is the portion that has, or can be written with, the phrase then in front of it.
Again, do not include the word then when asked to identify the conclusion.
If today is Tuesday, then tomorrow is Wednesday.

8. Converse
The converse of a conditional statement is formed by switching the hypothesis and conclusion.
If today is Tuesday, then tomorrow is Wednesday.
If tomorrow is Wednesday,
then today is Tuesday

9. Negation The negation is the opposite of the original statement.
Make the statement negative of what it was.
Use phrases like
Not, no, un, never, can’t, will not, nor, wouldn’t, etc.
Today is Tuesday.
Today is not Tuesday.

10. Inverse
The inverse is found by negating the hypothesis and the conclusion.
Notice the order remains the same!
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday,
then tomorrow is not Wednesday.

11. Contrapositive
The contrapositive is formed by switching the order and making both negative.
If today is Tuesday, then tomorrow is Wednesday.
If today is not Tuesday,
then tomorrow is not Wednesday.
If tomorrow is not Wednesday,
then today is not Tuesday.

12. Point, Line, Plane Postulates: Postulate 5
Through any two points there exists exactly one line. Y O

13. Point, Line, Plane Postulates: Postulate 6
A line contains at least two points.
Taking Postulate 5 and Postulate 6 together tells you that all you need is two points to make one line. H I

14. Point, Line, Plane Postulates: Postulate 7
If two lines intersect, then their intersection is exactly one point. B

15. Point, Line, Plane Postulates: Postulate 8
Through any three noncollinear points there exists exactly one plane. M R L

16. Point, Line, Plane Postulates: Postulate 9
A plane contains at least three noncollinear points.
Take Postulate 8 with Postulate 9 and this says you only need three points to make a plane. M R L

17. Point, Line, Plane Postulates: Postulate 10
If two points lie in a plane, then the line containing them lies in the same plane. M E

18. Point, Line, Plane Postulates: Postulate 11
If two planes intersect, then their intersection is a line.
Imagine that the walls of the classroom are different planes.
Ask yourself where do they intersect?
And what geometric figure do they form?

19. Homework 2.1 In Class Homework Due Tomorrow 1-8 10-50 ev, 51, 55, 56
p75-78
Homework
10-50 ev, 51, 55, 56
Due Tomorrow

20. Definitions and Biconditional Statements
Lesson 2.2
Definitions
and
Biconditional Statements

21. Lesson 2.2 Objectives Recognize a definition
Recognize a biconditional statement
Verify definitions using biconditional statements

22. Perpendicular Lines
Perpendicular lines intersect to form a right angle.
When writing that lines are perpendicular, we place a special symbol between the line segments
AB CD T

23. Definition The previous slide was an example of a definition.
It can be read forwards or backwards and maintain truth.

24. Biconditional Statement
A biconditional statement is a statement that is written, or can be written, with the phrase if and only if.
If and only if can be written shorthand by iff.
Writing a biconditional is equivalent to writing a conditional and its converse.
All definitions are biconditional statements.

25. Finding Counterexamples
To find a counterexample, use the following method
Assume that the hypothesis is TRUE.
Find any example that would make the conclusion FALSE.
For a biconditional statement, you must prove that both the original conditional statement has no counterexamples and that its converse has no counterexamples.
If either of them have a counterexample, then the whole thing is FALSE.

26. Example 1 If a+b is even, then both a and b must be even.
Assume that the hypothesis is TRUE.
So pick a number that is even (larger than 2)
Find any example that would make the conclusion FALSE.
Pick two numbers that are not even but add to equal the even number from above.
Those two numbers you picked are your counterexample.
If no counterexample can be found, then the statement is true.

27. Homework 2.2
In Class
3-12
p82-85
Homework
14-42 even
Due Tomorrow

28. Lesson 2.3
Deductive Reasoning

29. Lesson 2.3 Objectives
Use symbolic notation to represent conditional statements
Identify the symbol for negation
Utilize the Law of Detachment to form conclusions
Utilize the Law of Syllogism to form conclusions

30. Symbolic Conditional Statements
To represent the hypothesis symbolically, we use the letter p.
We are applying algebra to logic by representing entire phrases using the letter p.
To represent the conclusion, we use the letter q.
To represent the phrase if…then, we use an arrow, .
To represent the phrase if and only if, we use a two headed arrow,

31. Example of Symbolic Representation
If today is Tuesday, then tomorrow is Wednesday.
p =
Today is Tuesday
q =
Tomorrow is Wednesday
Symbolic form
p  q
We read it to say “If p then q.”

32. Negation Recall that negation makes the statement “negative.”
That is done by inserting the words not, nor, or, neither, etc.
The symbol is much like a negative sign but slightly altered… ~

33. Symbolic Variations Converse Inverse Contrapositive Biconditional
q  p
Inverse
~p  ~q
Contrapositive
~q  ~p
Biconditional
p q

34. Logical Argument
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument.
So deductive reasoning either states laws and/or conditional statements that can be written in if…then form.
There are two laws that govern deductive reasoning.
If the logical argument follows one of those laws, then it is said to be valid, or true.

35. Law of Detachment
If pq is a true conditional statement and p is true, then q is true.
It should be stated to you that pq is true.
Then it will describe that p happened.
So you can assume that q is going to happen also.
This law is best recognized when you are told that the hypothesis of the conditional statement happened.

36. Example 2
If you get a D- or above in Geometry, then you will get credit for the class.
Your final grade is a D.
Therefore…
You will get credit for this class!

37. Law of Syllogism
If pq and qr are true conditional statements, then pr is true.
This is like combining two conditional statements into one conditional statement.
The new conditional statement is found by taking the hypothesis of the first conditional and using the conclusion of the second.
This law is best recognized when multiple conditional statements are given to you and they share alike phrases.

38. Example 3 If tomorrow is Wednesday, then the day after is Thursday.
If the day after is Thursday, then there is a quiz on Thursday.
Therefore…
And this gets phrased using another conditional statement
If tomorrow is Wednesday, then there is a quiz on Thursday.

39. Deductive v Inductive Reasoning
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a proof.
This is often called a logical argument.
Inductive reasoning uses patterns of a sample population to predict the behavior of the entire population
This involves making conjectures based on observations of the sample population to describe the entire population.

40. Equivalent Statements
If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements!
Equivalent Statements
Conditional
Converse
Inverse
Contrapositive
If p, then q
If q, then p
If ~p, then ~q
If ~q, then ~p
Written just as it shows in the problem.
Switch the hypothesis with the conclusion.
Take the original conditional statement and make both parts negative.
Take the converse and make both parts negative.
If the converse is true, then the inverse is also true. Therefore they are equivalent statements!
Means “not”

41. Homework
In Class
1-5
p91-94
Homework
8-48 even
Due Tomorrow

42. Reasoning with Properties of Algebra
Lesson 2.4
Reasoning with
Properties of
Algebra

43. Lesson 2.4 Objectives Use properties from algebra to create a proof
Utilize properties of length and measure to justify segment and angle relationships

44. Algebraic Properties of Equality
Property
Definition
Identification
Abbreviation
Addition
If a=b, then a+c = b+c.
Something is added to both sides of the equation.
APOE
Subtraction
If a=b, then a-c = b-c.
Something is subtracted from both sides of the equation.
SPOE
Multiplication
If a=b, then ac = bc.
Something is multiplied to both sides of the equation.
MPOE
Division
If a=b and c≠0, then
a/c = b/c.
Something is being divided into both sides.
DPOE
Substitution
If a=b, then a can be substituted for b in any expression.
One object is used in place of another without any calculations being done.
SUB
Distributive
a(b+c) = ab + ac
A number outside of parentheses has been multiplied to all numbers inside.
DIST

45. Reflexive, Symmetric and Transitive Properties
Definition
For any real number a, a = a
If a=b, then b=a.
If a=b and b=c, then a=c.
How to
Remember
Reflexive is close to reflection, which is what you see when you look in a mirror.
Symmetric starts with s, so that means to switch the order.
Transitive is like transition, and when a and c equal the same thing, they must transition to equal each other.
How to Use
This will be used when two objects share something, such as sharing a common side of a triangle
This is a step that allows you to change the order of objects so they fit where you need them.
This is used most often in proofs, and can be often thought of as substitution.

46. Show Your Work This section is an introduction to proofs.
To solve any algebra problem, you now need to show ALL steps.
And with those steps you need to give a reason, or law, that allows you to make that step.
Remember to list your first step by simply rewriting the problem.
This is to signify how the problem started.

47. Example 4 Solve 9x+18=72 9x+18=72 Given 9x=54 SPOE x=6 DPOE -18 9
Short for “Information given to us.”
9x+18=72
Given
-18
9x=54
SPOE 9
x=6
DPOE

48. Example 5: Using Segments
In the diagram, AB=CD. Show that AC=BD. A B C D
Think about changing AB into AC? And the same with CD into BD?
AB=CD
Given
AB+BC=BC+CD
APOE
Segment Addition
Postulate
AC=AB+BC
Segment Addition
Postulate
BD=BC+CD
AC=BD
Transitive POE

49. Example 6: Using Angles HW Problem #24, p100
In the diagram, m RPQ=m RPS, verify to show that m SPQ=2(m RPQ).
mRPQ=m RPS
Given
Angle Addition
Postulate
m SPQ=m RPQ+m RPS S R Q P
m SPQ=m RPQ+m RPQ
SUB
m SPQ=2(m RPQ)
DIST

50. Example 7
Fill in the two-column proof with the appropriate reasons for each step
APOE
MPOE
Symmetric POE

51. Homework 2.4 In Class Homework Due Tomorrow 1,4-8 10-32, 36-50 even
p99-101
Homework
10-32, even
Due Tomorrow

52. Proving Statements about Segments
Lesson 2.5
Proving Statements about Segments

53. Lesson 2.5 Objectives Write a two-column proof
Justify statements about congruent segments

54. Theorem
A theorem is a true statement that follows the truth of other statements.
Theorems are derived from postulates, definitions, and other theorems.
All theorems must be proved.

55. Two-Column Proof
One method of proving a theorem is to use a two-column proof.
A two-column proof has numbered statements and corresponding reasons placed in a logical order.
That logical order is just steps to follow much like reading a cook book.
The first step in a two-column proof should always be rewriting the information given to you in the problem.
When you write your reason for this step, you say “Given”.
The last step in a two-column proof is the exact statement that you are asked to show.

56. Example 8 Prove the Symmetric Property of Segment Congruence.
GIVEN: Segment PQ is congruent to Segment XY
PROVE: Segment XY is congruent to Segment PQ

57. Hints for Making Proofs
Remember to always write down the first step as given information.
Develop a mental plan of how you want to change the first statement to look like the last statement.
Try to evaluate how you can make each step change from the previous by applying some rule.
You must follow the postulates, definitions, and theorems that you already know.
Number your steps so the statements and the reasons match up!

58. Example 9
Fill in the missing steps
Transitive POE
A  C

59. Example 10 Fill in the missing steps 1 and 2 are a linear pair
1 and 2 are supplementary
Definition of supplementary angles
m1 = 180o - m2

60. Homework 2.5 In Class Homework Due Tomorrow 1,3-5,7,9 6-11,16,21,22p
Homework
6-11,16,21,22
Due Tomorrow

61. Proving Statements about Angles
Lesson 2.6
Proving Statements about Angles

62. Lesson 2.6 Objectives
Utilize the angle and segment congruence properties
Prove properties about special angle pairs

63. Theorem 2.1: Properties of Segment Congruence
Segment congruence is always
Reflexive
Segment AB is congruent to Segment AB.
Symmetric
If AB  CD, then CD  AB.
Transitive
If AB  CD and CD  EF, then AB  EF.

64. Theorem 2.2: Properties of Angle Congruence
Angle congruence is always
Reflexive
A  A
Symmetric
If A  B, then B  A.
Transititve
If A  B and B  C, then A  C.

65. Theorem 2.3: Right Angle Congruence Theorem
All right angles are congruent. 1 2
GIVEN: 1 and 2 are right angles.
PROVE: 1  2
1. 1 and 2 are right angles
1. Given
2. m1 = 90o, m2 = 90o
2. Definition of Right Angles
3. m1 = m2
3. Trans POE
4. 1  2
4. DEFCON

66. Theorem 2.4: Congruent Supplements Theorem
If two angles are supplementary to the same angle, or congruent angles, then they are congruent.
If m1 + 2 = 180o and m2 + m3 = 180o, then 1  3. 2 1 3

67. Theorem 2.5: Congruent Complements Theorem
If two angles are complementary to the same angle, or to congruent angles, then they are congruent.
If m4 + m5 = 90o and m5 + m6 =90o, then 4  6. 5 4 6

68. Postulate 12: Linear Pair Postulate
The Linear Pair Postulate says if two angles form a linear pair, then they are supplementary.
1 + 2 = 180o 1 2

69. Theorem 2.6: Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Vertical angles are angles formed by the intersection of two straight lines.
1  3 2 1 3 4
2  4

70. Example 11
Using the following figure, fill in the missing steps to the proof.
Given 2
Definition of a linear pair 4
m1 + m2 = 180o
m3 + m4 = 180o
Congruent Supplements Theorem

71. Homework 2.6 In Class Homework Due Tomorrow 1,3-9,10,23p
Homework
10, 12-22, 27-28, 33-36
Due Tomorrow