– Justifications and Proofs

Measuring, describing, and transforming: these are three major skills in geometry that you have been developing. In this chapter, you will focus on comparing; you will explore ways to determine if two figures have the same shape (called similar) and if they have the same size (congruent).

Making logical and convincing arguments that support specific ideas about the shape you are studying is another important skill. In this chapter you will learn how you can document facts to support a conclusion in a flowchart and two-column proof.

In this chapter, you will learn:• how to support a mathematical statement using flowcharts and two-column proofs

• what a converse of a conditional statement is and how to recognize whether or not the converse is true

• how to disprove a statement with a counterexample

• about the special relationships between shapes that are similar or congruent

• how to determine if triangles are similar and/or congruent

4.1

How Can I Measure An Object?

Pg. 3Units of Measure

4.1 – What Is A Proof?Types of Proof

Whenever you buy a new product that needs to be put together, you are given a set of directions. The directions are written in a specific order that must be followed closely in order to get the desired finished product. Sometimes they clarify their directions by explaining why you are completing each step. This is the same idea we use in geometry in proofs.

4.1 – ORDERING STATEMENTS When you write a proof, the statements must be in a specific order, building off of each other. You can't just jump to the end without breaking down each part. To illustrate this, with your group explain how to make a peanut butter and jelly sandwich. Work with your team to include all steps to make sure the sandwich will be made correctly.

4.2 – STATEMENTS AND REASONSWhen you write a proof in geometry, each statement you make must have a reason to support it. This helps people understand why each statement was listed. This can be done in a flowchart proof or a two-column proof. Examine the two types below. Notice where the statements and reasons are. Also, notice how the statements are in a specific order.

4.3 – REASONSThe reasons for certain statements come from definitions, properties, postulates, and theorems. Below are some commonly used reasons.

Name Property of Equality

Addition Property

If a = b, then

Subtraction Property

If a = b, then

Multiplication Property

If a = b, then

Division Property

If a = b, then

Distribution Property

If a(x + b), then

a + c = b + c

a - c = b - c

ac = bc

a/c = b/c

ax + ab

Substitution Property

If a = b, then

Reflexive Property

Transitive Property

If a = b and b = c, then

b can replace a

a = a

a = c

a. Write the reason for each statement

Statement Reason

If 4(x + 7), then 4x + 28

If 2x + 5 = 9, then 2x = 4

If x – 7 = 2, then x = 9

If 4x = 12, then x = 3

If a = 20, then 5a = 5(20)

If and , then X Y Y Z X Z

BD BD

Distributive Prop

Reflexive Prop

Subtraction Prop

Transitive Prop

Addition Prop

Division prop.

substitution

Statements Reasons

1. 5(x – 3) = 4(x + 2) 1. __________________

2. 5x – 15 = 4x + 8 2. _________________

3. x – 15 = 8 3. __________________

4. x = 23 4. __________________

c. write a reason for each statement.

given

Distributive Prop

Subtraction Prop

Addition Prop

GIVEN: M is the midpoint of , AM = 6inPROVE: AB = 12in

AB

Statements Reasons

1. M is the midpoint of 1. ___________________

2. AM = MB 2. ___________________

3. AM = 6in 3. ___________________

4. AM + MB = AB 4. ___________________

5. AM + AM = AB 5. ___________________

6. 6 + 6 = AB 6. ___________________

7. AB = 12in 7. ___________________

AB

given

Def. of midptgivenSegment Add. Post.SubstitutionSubstitutionSimplify

GIVEN: bisects ABC, mABD = 20°PROVE: ABC = 40°

BD

Statements Reasons

1. 1. bisects ABC Given

2. 2.mABD = mDBC Def. of angle bisector

3. 3.mABD = 20° given

4. 4.ABC = ABD + DBC Angle Addition Prop.

BD

5. 5.ABC = ABD + ABD Substitution

6. 6.ABC = 20° + 20° Substitution

7. 7.ABC = 40° Simplify

Statements Reasons

1. intersects 1. _______________________

2. 2. _______________________

3. 3. _______________________

4. 4. _______________________

5. 5. _______________________

6. 6. _______________________

CDAB given

Supplementary angles

Supplementary angles

Transitive prop.

subtraction

Def. of congruence

given

alt. interior =supplementary

1 + 3 = 180given

1 + 4 = 180 Def. of supplementaryTransitive prop.

3 = 4p // r Alt. interior =

MATH is a parallelogram givenOpp. sides of parallelogram =

given

MH MN Transitive prop.

Base in isos. ∆ =

Def. of congruence