Section 2-4: Reasoning in AlgebraTPI 32A: apply reflective, transitive, or symmetric prooperties 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• 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

Using Properties to Justify Steps in Solving Equations

Algebra Solve for x 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

3x = 129 Subtraction Property of Equality

x = 43 Division Property of Equality

Using Properties to Justify Steps in Solving Equations

Solve for x 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

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.

Reflective Property of Congruence

AB ABA 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

Using Properties of Equality and Congruence

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

a. K K Reflective Property of

Symmetric Property of Equality

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

c. If RS TW and TW PQ,

then RS PQ.

Transitive Property of

d. If m A = mB, then

m B = mA

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