Chapter 2

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-then form. This is a logical statement that contains

two parts• Hypothesis• Conclusion

If today is Tuesday, then tomorrow is Wednesday.

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.


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.


Converse The converse of a conditional

statement is formed by switching the hypothesis and conclusion.

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 not Tuesday,


then tomorrow is not Wednesday.

Contrapositive The contrapositive is formed by

switching the order and making both negative.

If tomorrow is not Wednesday,


If today is not Tuesday, then tomorrow is not Wednesday.

then today is not Tuesday.

Point, Line, Plane Postulates:Postulate 5

Through any two points there exists exactly one line.

Y O

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


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

B


Through any three noncollinear points there exists exactly one plane.

M

R

L

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


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

M E

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

DefinitionsandBiconditional 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

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.

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.


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


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 argument follows one of those laws, then it is said to be valid, or true.

Law of Detachment If pq is a true conditional statement and

p is true, then q is true. It should be stated to you that pq 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.

Your final grade is a D. Therefore…

You will get credit for this class!

Law of Syllogism If pq and qr are true conditional

statements, then pr 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 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.

Equivalent StatementsConditiona

lConverse 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.

Means “not”

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

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

Homework In Class

1-5• p91-94

Homework 8-48 even

Due Tomorrow

Lesson 2.4

Reasoning withProperties ofAlgebra

Lesson 2.4 Objectives Use properties from algebra to

create a proof Utilize properties of length and

measure to justify segment and angle relationships

Algebraic Properties of Equality

Property Definition Identification Abbreviation

AdditionProperty

If a=b, then a+c = b+c.Something is added to both sides of the equation. APOE

SubtractionProperty

If a=b, then a-c = b-c.Something is subtracted from both sides of the equation.

SPOE

MultiplicationProperty

If a=b, then ac = bc.Something is multiplied to both sides of the equation. MPOE

DivisionProperty

If a=b and c≠0, thena/c = b/c.

Something is being divided into both sides. DPOE

SubstitutionProperty

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

DistributiveProperty

a(b+c) = ab + ac

A number outside of parentheses has been multiplied to all numbers inside.

DIST

Reflexive, Symmetric and Transitive Properties

Reflexive Symmetric Transitive

DefinitionFor any real number a,

a = a

If a=b, then b=a.

If a=b and b=c, then a=c.

Howto

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.

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.

Example 4Solve 9x+18=72

9x+18=72 GivenShort for “Information given to us.”

9x=54

x=6

-18 -18

SPOE

DPOE

9 9

Example 5: Using SegmentsIn the diagram, AB=CD. Show that AC=BD.

A B C D

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

AB+BC=BC+CD

AC=AB+BC

BD=BC+CD

AC=BD

APOE

Segment AdditionPostulate

Transitive POE

Segment AdditionPostulate

Example 6: Using Angles

HW Problem #24, p100 In the diagram, m RPQ=m RPS, verify to

show that m SPQ=2(m RPQ).

mRPQ=m RPS Given

m SPQ=m RPQ+m RPS

m SPQ=m RPQ+m RPQ

m SPQ=2(m RPQ)

Angle AdditionPostulate

SUB

DIST

S

R

Q

P

Example 7Fill in the two-column proof with the appropriate reasons for each step

APOE

MPOE

Symmetric POE


1,4-8• p99-101

Homework 10-32, 36-50 even

Due Tomorrow

Lesson 2.5

Proving Statements about Segments

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. Theorems are 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-column proof 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 9Fill in the missing steps

Transitive POE

A C

Example 10Fill in the missing steps

1 and 2 are a linear pair

1 and 2 are supplementary

Definition of supplementary angles

m1 = 180o - m2


1,3-5,7,9• p105-107

Homework 6-11,16,21,22

Due Tomorrow

Lesson 2.6

Proving Statements about Angles

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.

Theorem 2.3:Right Angle Congruence Theorem

All right angles are congruent.

GIVEN: 1 and 2 are right angles.PROVE: 1 2

1. 1 and 2 are right angles 1. Given

2. m1 = 90o, m2 = 90o 2. Definition of Right Angles

3. m1 = m2 3. Trans POE

4. 1 2 4. DEFCON

2

1

Theorem 2.4:Congruent Supplements Theorem

If two angles are supplementary to the same angle, or congruent angles, then they are congruent. If m1 + 2 = 180o and m2 + m3 = 180o,

then 1 3.

12

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 m4 + m5 = 90o and m5 + m6 =90o,

then 4 6.

54

6

Postulate 12:Linear Pair Postulate

The Linear Pair Postulate says if two angles form a linear pair, then they are supplementary.

1 2

1 + 2 = 180o

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.

12

34

1 3

2 4

Example 11Using the following figure, fill in the

missing steps to the proof.

Given

2

4

Definition of a linear pair

m1 + m2 = 180o

m3 + m4 = 180o

Congruent Supplements Theorem


1,3-9,10,23• p112-116

Homework 10, 12-22, 27-28, 33-36

Due Tomorrow

Chapter 2

Documents

plane postulates

conditional statementslesson

noncollinear points

original statement

logical statement

hypothesisthe hypothesis

conclusionthe conclusion

different planes