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Chapter 2. Reasoning and Proof. 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

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Chapter 2

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## Chapter 2

Reasoning and Proof

### 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

## Lesson 2.1

Conditional Statements

### Lesson 2.1 Objectives

• Analyze conditional statements

• Write postulates about points, lines, and planes using conditional statements

### Conditional Statements

• A conditional statement is any statement that is written, or can be written, in the if-thenform.

• This is a logical statement that contains two parts

• Hypothesis

• Conclusion

If today is Tuesday, then tomorrow is Wednesday.

### Hypothesis

• The hypothesisof a conditional statement is the portion that has, or can be written, with the word ifin front.

• When asked to identify the hypothesis, you do not include the word if.

If today is Tuesday, then tomorrow is Wednesday.

### 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.

### Converse

• The converseof 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

### 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.

### 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.

### 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.

Iftomorrow is not Wednesday,

then today is not Tuesday.

Y

O

### Point, Line, Plane Postulates:Postulate 5

• Through any two points there exists exactly one line.

H

I

### Point, Line, Plane Postulates:Postulate 6

• A line contains at leasttwo points.

• Taking Postulate 5 and Postulate 6 together tells you that all you need is two points to make one line.

B

### Point, Line, Plane Postulates:Postulate 7

• If two lines intersect, then their intersection is exactly one point.

R

L

M

### Point, Line, Plane Postulates:Postulate 8

• Through any three noncollinear points there exists exactly one plane.

R

L

M

### 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

E

### Point, Line, Plane Postulates:Postulate 10

• If two points lie in a plane, then the line containing them lies in the same plane.

### 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?

### Homework 2.1

• In Class

• 1-8

• p75-78

• Homework

• 10-50 ev, 51, 55, 56

• Due Tomorrow

## Lesson 2.2

Definitions

and

Biconditional Statements

### Lesson 2.2 Objectives

• Recognize a definition

• Recognize a biconditional statement

• Verify definitions using biconditional statements

### 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

### Definition

• The previous slide was an example of a definition.

• It can be read forwards or backwards and maintain truth.

### 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.

### Finding Counterexamples

• To find a counterexample, use the following method

• Assume that the hypothesis is TRUE.

• Find any example that would make the conclusionFALSE.

• 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.

### 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 conclusionFALSE.

• 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.

• In Class

• 3-12

• p82-85

• Homework

• 14-42 even

• Due Tomorrow

## Lesson 2.3

Deductive Reasoning

### 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

### 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, .

### 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.”

### 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…

• ~

### Symbolic Variations

• Converse

• q  p

• Inverse

• ~p  ~q

• Contrapositive

• ~q  ~p

• Biconditional

• p q

### 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 argumentfollows one of those laws, then it is said to be valid, or true.

### 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.

### Example 2

• If you get a D- or above in Geometry, then you will get credit for the class.

• Therefore…

• You will get credit for this class!

### 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.

### 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.

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.

### Deductive v Inductive Reasoning

If the conditional statement is true, then the contrapositive is also true. Therefore they are equivalent statements!

### Equivalent Statements

Means “not”

If the converse is true, then the inverse is also true. Therefore they are equivalent statements!

• In Class

• 1-5

• p91-94

• Homework

• 8-48 even

• Due Tomorrow

## Lesson 2.4

Reasoning with

Properties of

Algebra

### Lesson 2.4 Objectives

• Use properties from algebra to create a proof

• Utilize properties of length and measure to justify segment and angle relationships

### Reflexive, Symmetric and Transitive Properties

• 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.

-18

9

-18

9

### Example 4

Solve 9x+18=72

Short for “Information given to us.”

Given

9x+18=72

SPOE

9x=54

DPOE

x=6

D

C

B

A

### Example 5: Using Segments

In the diagram, AB=CD. Show that AC=BD.

Think about changing AB into AC? And the same with CD into BD?

Given

AB=CD

AB+BC=BC+CD

APOE

Postulate

AC=AB+BC

Postulate

BD=BC+CD

Transitive POE

AC=BD

S

R

P

Q

### 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

Postulate

m SPQ=m RPQ+m RPS

SUB

m SPQ=m RPQ+m RPQ

DIST

m SPQ=2(m RPQ)

### Example 7

Fill in the two-column proof with the appropriate reasons for each step

APOE

MPOE

Symmetric POE

### Homework 2.4

• In Class

• 1,4-8

• p99-101

• Homework

• 10-32, 36-50 even

• Due Tomorrow

## Lesson 2.5

### Lesson 2.5 Objectives

• Write a two-column proof

• Justify statements about congruent segments

### Theorem

• A theorem is a true statement that follows the truth of other statements.

• Theoremsare derived from postulates, definitions, and other theorems.

• All theorems must be proved.

### 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-columnproof is the exact statement that you are asked to show.

### 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

### 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!

### Example 9

Fill in the missing steps

Transitive POE

A C

### 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

### Homework 2.5

• In Class

• 1,3-5,7,9

• p105-107

• Homework

• 6-11,16,21,22

• Due Tomorrow

## Lesson 2.6

### Lesson 2.6 Objectives

• Utilize the angle and segment congruence properties

• Prove properties about special angle pairs

### 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.

### 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.

1

2

### Theorem 2.3:Right Angle Congruence Theorem

• All right angles are congruent.

GIVEN:1 and 2 are right angles.

PROVE:1 2

2.Definition of Right Angles

2.m1 = 90o, m2 = 90o

3.Trans POE

3.m1 = m2

4.1 2

4.DEFCON

### 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

### 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

### 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

### Theorem 2.6:Vertical Angles Theorem

• If two angles are vertical angles, then they are congruent.

• Vertical anglesare angles formed by the intersection of two straight lines.

2

1 3

1

3

4

2 4

### Example 11

Using the following figure, fill in the missing steps to the proof.

Given

Definition of a linear pair

2

4

m1 + m2 = 180o

m3 + m4 = 180o

Congruent Supplements Theorem

### Homework 2.6

• In Class

• 1,3-9,10,23

• p112-116

• Homework

• 10, 12-22, 27-28, 33-36

• Due Tomorrow