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2.5:Reason Using Properties from Algebra

Goal:Use algebraic properties in logical arguments.

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Algebraic Properties of Equality• Let a, b, and c be real numbers.

Addition Property If , then .a cb c ba

Subtraction Property If , then .a cb c ba

Multiplication Property If , then .ac ca b b

Division Property If and 0, then .a b ca b

c c

Substitution Property If , then n be ca aa b substituted fo r y in anb

equation or expression.

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Example 1:Write reasons for each stepSolve 2 3 9 . Write a reason for each step.

Equation Explanation Reason

x x

2 3 9 Write Given

original

equa io

t n.

x x

2 3 9 Add to

each sid

Addition

Property

e.

of Equality

x xx x x

3 9 Simplif3 Combine

lik

y

ms

e ter

x

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Example 1:Write reasons for each step

3 6 Subtraction

3 Property o

Subtract

from

each sid

e

f

E a qu l

x

ity

Divide each

2 Division

3 Propert

y o f

side by

x

Equality

the value of 2is .x

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Distributive Property

• a(b + c) = ab + acab + ac, where a, b, and c are real numbers.

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Example 2:Use the Distributive PropertySolve - 4(6 2) 64. Write a reason for each step.x

Equation Explanation Reason

4(6 2) 64 Write original Given

equation.

x

24 8 Distributive

64

Multiply

Pro r y

pe t

x

Add to

each side.

Prope

24 72 8 Add

rty of E

itio

q ity

n

ual

x

Divide each

side by

.

Prope

3 Division

24 rty of Equality

x

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Checkpoint

Solve 5 7 2 . Write a reason for each step.x x

5 7 2 Givenx x 5 7 2 Subtraction Property

of Equality

x x x x

5 7 Simplifyx

12 Subtraction Property

of Equality

x

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Checkpoint

Solve 4(5 ) 2 . Write a reason for each step.x x

4(5 ) 2 Givenx x 20 4 2 Distributive Propertyx x

20 2 Addition Property of Equalityx 10 Division Property of Equalityx

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Example 3:Use properties in the real world• Speed:A motorist travels 5 miles per hour slower

than the speed limit s for 3.5 hours. The distance traveled d can be determined by the formula d = 3.5(s – 5). Solve for s.

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Example 3:Use properties in the real world

Equation Explanation Reason

3.5( 5) Write original Givend s

3.5 17.5 Distributive

Property

Multiply sd

Add

to eac

17.5 3.5 17.5 Additi

h side Property of Equ li

on

a ty

sd

Divide each

side by

17.5Divisio

Property o

n3.

f E

5

3.5 quality

ds

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Reflexive Property of Equality

Real numbers For any real number , .a a a

Segment Length For any segment , .A ABB AB

Angle Measure For any angle , .m A AA m

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Symmetric Property of EqualityReal numbers For any real numbers and , if ,

then .

a b b

b a

a

Segment Length For any segments and , if

, then .

AB CD

AB CD CD AB

Angle Measure For any angles and , if ,

then .

A B m A

m B

m

m A

B

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Transitive Property of EqualityReal numbers For any real numbers , , and ,

if and , then .

a b c

a b b c a c

Segment Length For any segments , , and , if

and , then .

AB CD EF

AB CD CD AB EEF F

Angle Measure For any angles , , and , if

and ,

then .

A B C

m A

m A m B m

m

m

C

B C

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Example 4:Use properties of equality

Show that .CF AD

Equation Reason

Given

G iven

EFAB

B EC D

Segment Addition

Postulate

AC AB BC

Segment Addition Postul a teDE EFDF

Property of

Subs

ti

Equali

tution

ty

DF BC AB

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Example 4:Use properties of equality

Tran Property ositive f EqualityCDF A

Addition Property of EqualityACDF CD CD

Substitution Property of EqualityCF AD

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Checkpoint

5( 3) Givend s

5 15 Distribu tive Propert y d s 15 5 Subtraction Property of Equalityd s 15

Division Property of Equality5

ds

Suppose the equation in Example 3 is 5( 3).

Solve for . Write a reason for each step.

d s

s

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Checkpoint:Name the property of equality

If , then .GH JK JK GH

Symmetric Property of Equality for Segment

Length

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Checkpoint:Name the property of equality

If , and 44, then 44.r s s r

Transitive Property of Equality for Real Numbers

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Checkpoint:Name the property of equality

.m N m N

Reflexive Property of Equality for Angle

Measure