3.6

What is a Proof?

Pg. 19Types of Proof

3.6 – 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.

3.31 – 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.

3.32 – 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

Symmetric PropertyTransitive Property

If a = b and b = c, then

b can replace a

a = a

a = c

If a = b, then b = a

1. Use the property to complete the statement.

3 2m

5(20) 100

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 and , then A B B C A C

BD BD

Distributive Prop.

Reflexive Prop.

Subtraction Prop.

Transitive Prop.

Addition Prop.

Symmetric

Division Prop.

PROOFS!All proofs start and end the same. It is very helpful to have a picture to refer to. We are given information and then are told to prove something.

Given:Prove:

Picture

Statements Reasons

Format of Proofs

1.2.3.…

1.2.3.…

What you are given Given

To Prove

Complete the logical argument by writing a reason for each step.

Statements Reasons

1. 8x – 34 = 6 1. __________________

2. 8x = 40 2. __________________

3. x = 5 3. __________________

given

Addition Prop

Division Prop

Complete the logical argument by writing a reason for each step.

Statements Reasons

1. 4x – 7 = 6x + 7 1. __________________

2. -2x – 7 = 7 2. __________________

3. -2x = 14 3. __________________

4. x = -7 4. __________________

given

Subtraction Prop

Addition Prop

Division Prop

Complete the logical argument by writing a reason for each step.

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. _______________________

given

Distributive Prop

Subtraction Prop

Addition Prop

4. Solve the equation. Write a reason for each step 4x – 9 = 2x + 11

Statements Reasons

1. 1.4x – 9 = 2x + 11 Given

2. 2.2x – 9 = 11 Subtraction Prop.

3. 3.2x = 20 Addition Prop.

4. 4.x = 10 Division Prop.

3.33 Solve the equation for y. Write a reason for each step.

12x – 3y = 30

Statements Reasons

1. 1.12x – 3y = 30 Given

2. 2. –3y = -12x + 30 Subtraction Prop.

3. 3.y = 4x – 10 Division Prop.

3.34 – GEOMETRIC PROOFComplete the proof by writing a reason for each step.

GIVEN: AL = SKPROVE: AS = LK

C. Given

E. Reflexive

F. Addition Prop.

A. Segment Addition

D. Segment Addition

B. Substitution

givenDef. of MidpointgivenSegment Addition

SubstitutionSubstitutionSimplify

givenDef. Segment BisectorgivenSegment Addition

SubstitutionSubstitutionSimplify

Statements Reasons

1. 1. Given

2. 2. Def. of angle bisector

3. 3. given

4. 4. Angle Addition

5. 5. Substitution

6. 6. Substitution

7. 7. Simplify

mABD mDBCmABD 20

ABC ABD DBC

ABC ABD ABD

ABC 20 20ABC 40