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Algebraic Proof
Algebraic Proof
Uses:
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Number Properties
Definitions
Postulates
Properties of Equality

Addition Property:
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
if a = b, then a+c = b+c
Subtraction Property:
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
if a = b, then a-c = b-c
Multiplication Property:
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
if a = b, then ac = bc
Division Property:
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if a = b, then a/c = b/c
Properties of Equality

Reflexive Property:
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
a=a
Symmetric Property:
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
if a = b, then b = a
Transitive Property:
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if a = b and b = c, then a = c
Substitution Property:
if a = b, then b can be substituted for a
Definitions

Congruent Segments have the same length

Congruent Angles have the same measure

Right Angle is an angle measuring 90
degrees
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Complementary Angles add to 90 degrees

Supplementary Angles add to 180 degrees
Postulates
Segment Addition Postulate:
If B is between A and C,
then AB + BC = AC.
Angle Addition Postulate:
If the ray BD is in the interior of <ABC
then m<ABD + m<BDC = m<ABC
Two Column Proof
Given: abc
Prove: xyz
Statements
Reasons
abc
Given
Connecting logic
....
xyz
Solve the Equation 3x – 7 =8
and Justify each step
Statements
3x
= 15
Reasons
Given
Addition Property
x
=5
Division Property
3x – 7 =
8
Solve the Equation
2(3x – 5) +3 = 17
Statements
Reasons
2(3x – 5) +3 = 17 Given
6x -10 +3 = 17 Distributive Property
6x -7 = 17
6x
= 24
X =4
Simplify
Addition Property
Division Property
Theorems
Unlike a Postulate
Which we Assume to be True,
A Conjecture is a statement
We think is True.
Once it has been proven
It is called a Theorem
Using Theorems
Before we can
Use a Theorem
In a proof
We MUST first Prove it!
Right Angle Congruence Theorem
Given: <A and <B are Right Angles
Prove: <A is congruent to <B
Statements
Reasons
<A and <B are Right
Angles
m<A =90;
m<B =90
m<A = m<B
Given
<A is Congruent to <B
Definition of Congruent
Angles
Definition of Right
Angle
Substitution Property
Congruent Complements Theorem
Given: <A and <B are Complements to <C
Prove: <A is congruent to <B
Statements
Reasons
<A and <B are
Complements to <C
m<A +m<C=90;
m<B +m<C=90
m<A +m<C= m<B+m<C
Given
m<A
= m<B
<A is Congruent to <B
Subtraction Property
Definition of Congruent
Angles
Definition of
Complementary Angles
Substitution Property
Congruent Supplements Theorem
Given: <A and <B are Supplements to <C
Prove: <A is congruent to <B
Statements
Reasons
<A and <B are
Supplements to <C
m<A +m<C=180;
m<B +m<C=180
m<A +m<C= m<B+m<C
Given
m<A
= m<B
<A is Congruent to <B
Subtraction Property
Definition of Congruent
Angles
Definition of
Supplementary Angles
Substitution Property
Linear Pair Theorem
Given: <1 and <2 are a Linear Pair
Prove: <1 and <2 are Supplementary
Statements
Reasons
<1 and <2 are a
Linear Pair
m<1 +m<2=m<ABC
Given
Angle Addition
Postulate
m<ABC = 180
Definition of Straight
Angle
m<1+m<2 =180
Substitution Property
<1 & <2 Supplementary Definition of
Supplementary Angles