Download - 1 2.5:Reason Using Properties from Algebra Goal:Use algebraic properties in logical arguments
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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