Download Sec. 2-4 Reasoning in Algebra

Sec. 2-4
Reasoning in Algebra
Objective: To connect reasoning in
Algebra & Geometry
Objectives
Review properties of equality and use
them to write algebraic proofs.
Identify properties of equality and
congruence.
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In Geometry you accept postulates & properties
as true.
You use Deductive Reasoning to prove other
statements.
In Algebra you accept the Properties of Equality
as true also.
Algebra Properties of Equality
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Addition Property:
If a = b, then a + c = b + c
Subtraction Property:
If a = b, then a – c = b – c
Multiplication Property:
If a = b, then a • c = b • c
Division Property:
If a = b, then a/c = b/c (c ≠ 0)
More Algebra Properties
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Reflexive Property:
a = a (A number is equal to itself)
Symmetric Property:
If a = b, then b = a
Transitive Property:
If a = b & b = c, then a =c
2 more Algebra Properties
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Substitution Properties: (Subs.)
If a = b, then “b” can replace “a” anywhere
Distributive Properties:
a(b +c) = ab + ac
A proof is an argument that uses logic, definitions,
properties, and previously proven statements to show
that a conclusion is true.
An important part of writing a proof is giving
justifications to show that every step is valid.
Example 1: Algebra Proof
3x + 5 = 20
-5 -5
3x = 15
3
3
x=5
5=x
1. Given Statement
2. Subtr. Prop
3. Division Prop
4. Symmetric Prop
Example 2 :
 Addition Proof
A
Given: mAOC = 139
Prove: x = 43
Statements
1. mAOC = 139, mAOB = x,
mBOC = 2x + 10
2. mAOC = mAOB + mBOC
3.
4.
5.
6.
7.
139 = x + 2x + 10
139 = 3x + 10
129 = 3x
43 = x
x = 43
B
x
(2x + 10)
C
O
1.
2.
3.
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5.
6.
7.
Reasons
Given
 Addition Prop.
Subs. Prop.
Addition Prop
Subtr. Prop.
Division Prop.
Symmetric Prop.
Example 3: Segment Addition Proof
Given: AB = 4 + 2x
A
BC = 15 – x
4 + 2x
AC = 21
Prove: x = 2
1.
2.
3.
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5.
6.
Statements
AB=4+2x, BC=15 – x,
AC=21
AC = AB + BC
21 = 4 + 2x + 15 – x
21 = 19 + x
2=x
x=2
1.
2.
3.
4.
5.
6.
B
15 – x
C
Reasons
Given
Segment Add. Prop.
Subst. Prop.
Combined Like Term.
Subtr. Prop.
Symmetric Prop.
You learned in Chapter 1 that segments with
equal lengths are congruent and that angles with
equal measures are congruent. So the Reflexive,
Symmetric, and Transitive Properties of Equality
have corresponding properties of congruence.
Geometry Properties of Congruence
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Reflexive Property: AB  AB
A  A
Symmetric Prop: If AB  CD, then CD  AB
If A  B, then B  A
Transitive Prop:
If AB  CD and CD  EF, then AB  EF
IF A  B and B  C, then A  C
What did I learn Today?
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Name the property for each of the following
steps.
P  Q, then Q  P
Symmetric Prop
TU  XY and XY  AB, then TU  AB
Transitive Prop
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DF  DF
Reflexive