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2.5 Algebraic Proof

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2.5 Algebraic Proof

Objectives:Review properties of equality and use them to write algebraic proofs.Identify properties of equality and congruence.

Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

Section 2-5: Reasoning in AlgebraStandard: apply reflective, transitive, or symmetric properties of equality or congruence

Objectives:• Connect reasoning in algebra and geometry• Justify steps in deductive reasoning

In geometry • postulates, definitions, & properties are accepted as true (refer to page 842 for a complete list of postulates)• you use deductive reasoning to prove other statements

We will look at some basic properties used to justify statements…..

….. which leads to writing proofs.

Addition Property of Equality

If a = b, then a + c = b + c Add same amount to both sides of an equation.

Subtraction Property of Equality

If a = b, then a - c = b - c Subtract same amount to both sides of an equation.

Multiplication Property of Equality

If a = b, then a ∙ c = b ∙ c Multiply both sides of an equation by the same amount.

Division Property of Equality

If a = b and c 0, then

Divide both sides of an equation by the same amount.

a b

c c

Reflective Property of Equality

a = a Ex: 5 = 5

Symmetric Property of Equality

If a = b, then b = a Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same.

Transitive Property of Equality

If a = b and b = c, then a = c.EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2.

Substitution Property of Equality

If a = b , then b can replace a in any expression. Ex: a = 3; If a = b, then 3 = 3.

Distributive Property

a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9

2.5 Properties of EqualityTable on page #113

The Distributive Property states that

a(b + c) = ab + ac.

Remember!

Reflective Property of Congruence

AB AB A A

Symmetric Property of Congruence

Transitive Property of Congruence

If AB CD and AB EF, then CD EF.If A B and B C, then A C.

The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements.

If AB CD, then CD AB.If A B, then B A

2.5 Properties of CongruenceTable on page #114

What’s the Difference between equality and congruence?

A B

AB represents the length AB, so you can think of AB as a variable representing a number.

Helpful Hint

Congruence

Geometric objects (figures / drawings) can be congruent to each other.

Equality

Measurements (numbers)can be equal to each other.

Numbers are equal (=) and

figures are congruent ().

Remember!

Statements use symbolStatements use = symbol

2.5 ApplicationWrite a justification for each step.

NO = NM + MO Segment Addition Post.

4x – 4 = 2x + (3x – 9) Substitution Property of Equality

4x – 4 = 5x – 9 Simplify.

–4 = x – 9 Subtraction Property of Equality

5 = x Addition Property of Equality

The basic format of a two column proof: Page 115

Given - facts you are given to use. STARTING POINT

Prove – conclusion you need to reach.ENDING POINT

Proof Example: Problem 3 page 116

This is given This is what you are

asked to prove

This is how you plan to get from the given to the

prove.

Reasons

Application

Statement Reason

AB + BC = AC

2y + 3y – 9 = 21

5y – 9 = 21

5y = 30

y = 6

Segment addition postulate

Substitution

Combine like terms

Addition Property (add 9 to both sides)

Division property (divide both sides by 5)

GIVEN: PROVE: y = 6

Using Properties to Justify Steps in Solving Equations

Algebra: Prove x = 43 and justify each step.

Given: m AOC = 139

m AOC = 139 Given

M AOB + m BOC = m AOC Angle Addition Postulate

x + 2x + 10 = 139 Substitution Property

3x + 10 = 139 Simplify or combine like terms

3x = 129 Subtraction Property of Equality

x = 43 Division Property of Equality

Prove : x = 43

Statement Reasons

Using Properties to Justify Steps in Solving Equations

Prove x = 20 and justify each step.

Given: LM bisects KLN

LM bisects KLN Given

MLN = KLM

4x = 2x + 40

2x = 40

x = 20

Def of Angle Bisector

Substitution Property

Subtraction Property of Equality

Division Property of Equality

Prove: x = 20

Statement Reasons

Using Properties to Justify Steps in Solving Equations

Solve for y and justify each step

Given: AC = 21

AC = 21 Given

AB + BC = AC Segment Addition Postulate

2y + 3y - 9 = 21 Substitution Property

5y – 9 = 21 Simplify

5y = 30 Addition Property of Equality

y = 6 Division Property of Equality

Find AB and BC by substituting y = 6 into the expressions.

Prove : y = 6

Now you try

Statement Reasons

Using Properties of Equality and Congruence

Name the property of congruence or equality the justifies each statement.

a. K K Reflective Property of Congruence

Symmetric Property of Equality

b. If 2x – 8 = 10, then 2x = 18Addition Property of Equality

c. If RS TW and

TW PQ,

then RS PQ.

Transitive Property of Congruence

d. If m A = m B, then

m B = m A

Use what you know about transitive properties to complete the following:

The Transitive Property of Falling Dominoes:

If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall.

C

HOMEWORK

COMPLETE 2-5 PACKET

DUE THURSDAY NOV 1