What is a Proof? Pg. 19 Types of Proof 3.6 What is a Proof? Pg. 19 Types 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 REASONS When 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 – REASONS The reasons for certain statements come from definitions, properties, postulates, and theorems. Below are some commonly used reasons.

a + c = b + c a - c = b - c ac = bc a/c = b/c ax + ab Name Property of Equality Addition Property If a = b, then Subtraction Property Multiplication Property Division Property Distribution Property If a(x + b), then a + c = b + c a - c = b - c ac = bc a/c = b/c ax + ab

a = a If a = b, then b = a a = c b can replace a Substitution Property Reflexive Property Symmetric Property Transitive Property If a = b and b = c, then b can replace a a = a If a = b, then b = a a = c

1. Use the property to complete the statement.

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 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.

Format of Proofs Given: Picture Prove: Statements Reasons 1. 2. 3. … What you are given 1. 2. 3. … Given To Prove

given Addition Prop Division Prop Statements Reasons 1. 8x – 34 = 6 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

given Subtraction Prop Addition Prop Division Prop Statements Reasons 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

given Distributive Prop Subtraction Prop Addition 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. 4x – 9 = 2x + 11 1. Given 2. 2x – 9 = 11 2. Subtraction Prop. 3. 2x = 20 3. Addition Prop. 4. x = 10 4. Division Prop.

3.33 Solve the equation for y. Write a reason for each step. 12x – 3y = 30 Statements Reasons 1. 12x – 3y = 30 1. Given 2. –3y = -12x + 30 2. Subtraction Prop. 3. y = 4x – 10 3. Division Prop.

C. Given E. Reflexive F. Addition Prop. A. Segment Addition 3.34 – GEOMETRIC PROOF Complete the proof by writing a reason for each step. GIVEN: AL = SK PROVE: AS = LK C. Given E. Reflexive F. Addition Prop. A. Segment Addition D. Segment Addition B. Substitution

given Def. of Midpoint given Segment Addition Substitution Substitution Simplify

given Def. Segment Bisector given Segment Addition Substitution Substitution Simplify

1. 1. Given 2. 2. 3. 3. given 4. 4. Angle Addition 5. 5. Substitution 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