Properties of Algebra

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Properties of Algebra

• There are various properties from algebra that
  allow us to perform certain tasks.

• We review them now to refresh your memory on the
  process and terminology.

• We will also add a few new properties which you
  might not be familiar.

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The proof is in the pudding.
Indubitably.
Le pompt de pompt le solve de crime!"
Deductive Reasoning
Je solve le crime. Pompt de pompt pompt."
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Properties of Equality
Addition property of equality
If a b and c d, then a c b d.
restated
Example
If a b and c d, then a c b d.
a b 3 3 a 3 b 3
Euclid referred to this property as Equals
when added to equals are equal.
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Properties of Equality
Subtraction property of equality
If a b and c d, then a - c b - d.
restated
Example
If a b and c d, then a - c b - d.
a b 3 3 a - 3 b - 3
Euclid referred to this property as Equals
when subtracted from equals are equal.
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Properties of Equality
Multiplication property of equality
If a b and c d, then ac bd.
restated
Example
If a b and c d, then ac bd.
a b 3 3 3a 3b
Euclid referred to this property as Equals
when multiplied by equals are equal.
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Properties of Equality
Division property of equality
If a b and , then
restated
Example
If a b and c c, then
a b 3 3
Euclid referred to this property as Equals
when divided by equals are equal.
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Properties of Equality
Division property of equality
If a b and , then
restated
If a b and c c, then
Why must c not equal zero?
You are not allowed to divide by zero.
Numbers divided by zero are undefined.
Euclid referred to this property as Equals
when divided by equals are equal.
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Properties of Equality
Reflexive property of equality
a a
This is really obvious. Nevertheless, it needs a
name.
When you look into a mirror, you see your
reflection. Think of the equal sign as
a mirror. It might help you remember
the term.
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Properties of Equality
Symmetric property of equality
If a b, then b a.
This is really obvious. Nevertheless, it needs a
name.
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Properties of Equality
Transitive property of equality
If a b and b c, then a c.
This is really obvious. Nevertheless, it needs a
name.
It might be helpful to associate this concept
with traveling from LA to NYC with a stop over at
Chicago. The transfer of planes allows
you to reach your final destination.
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Properties of Congruence
Reflexive property of congruence
This is really obvious. Nevertheless, it needs a
name.
When you look into a mirror, you see your
reflection. Think of the equal sign as
a mirror. It might help you remember
the term.
YES
Didnt we say this before?
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Properties of Congruence
Symmetric property of Congurence
If , then .
If , then .
This is really obvious. Nevertheless, it needs a
name.
Symmetric is the same for equality and congruence.
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Properties of Congruence
Transitive property of Congruence
It might be helpful to associate this concept
with traveling from LA to NYC with a stop over at
Chicago. The transfer of planes allows
you to reach your final destination.
This is the same as in equality.
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Distributive Property
a( b c) ab ac
Implied multiplication
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Recognition of Properties 1
If AB CD , then
Definition of congruent segments.
If a b and b c, then a c.
Transitive property of equality.
If a b 10 and b 3, then a 3 10.
Substitution property of equality.
If then AB CD.
Definition of congruent segments.
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Recognition of Properties 2
If a b and x y , then a x b y.
x y
Addition property of equality.
If a b and x y , then a - x b - y.
-x -y
Subtraction property of equality.
If a 7, then a 3 10.
3 3
Addition property of equality.
Why?
I added 3 to both sides. Remember Euclid?
Equals when added to equals are equal.
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Recognition of Properties 3
If B is on line AC and AB BC ,
then b is the midpoint of
Definition of midpoint.
If A B , then A 3 B 3.
3 3
Addition property of equality.
switch sides
Symmetric property of equality.
Reflexive property of equality.
Mirror image
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Recognition of Properties 4
11( 4x 7) 44x 77
Distributive property.
If a b and b c and c 11, then a 11.
Transitive property of equality.
If a 11 , then a 3 8.
-3 -3
Subtraction property of equality.
If a b and c 12, then
c c
Division property of equality.
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Recognition of Properties 5
7 7
If , then a 42.
Multiplication property of equality.
If 8x 48, then x 6.
__ ___ 8 8
Division property of equality.
If 2y 7 11, then 2y 18.
7 7
Addition property of equality.
If then
switch sides
Symmetric property of equality.
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Recognition of Properties 6
If AB 30 and A 5 , then 5B 30.
Substitution property of equality.
If B is the midpoint of , then AB
BC.
Definition of midpoint.
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Proofs
You have been doing proofs all along in Algebra
I. When?
When you solved equations, you were actually
doing proofs algebraic proofs.
The major difference between equations and
geometric proofs is in the form.
7( x 2 ) 35
Solving a first degree equation with 1
variable.
7x 14 35
14 14
7x 21
7 7
x 3
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Proofs
The major difference between equations and
geometric proofs is in the form.
If 7( x 2 ) 35, then x 3.
Written as a conditional.
Reasons
Statements
7( x 2 ) 35
Given Information
Distributive Property
7x 14 35
14 14
Reflexive Property
7x 21
Subtraction Prop. Of Equality
7 7
Reflexive Property
x 3
Division Prop. Of Equality
The only difference is that the
reasons/justification for each step must be
written in geometry.
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Finish
Start
Note this is a lot of writing.
Statements
Reasons
You will need To abbreviate
Given
4 4
Reflexive Property
3x 4( 7 x )
Multiplication Prop. Of Equality
Distributive Property
3x 28 4x
4x 4x
Reflexive Property
Addition Prop. Of Equality
7x 28
7 7
Reflexive Property
x 4
Division Prop. Of Equality
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Finish
Start
Statements
Reasons
This is a lot less writing.
Given
4 4
Reflexive Prop.
3x 4( 7 x )
Mult. Prop. Of
Distr. Prop. Of
3x 28 4x
4x 4x
Reflexive Prop.
Prop. Of
7x 28
7 7
Reflexive Prop.
x 4
Div. Prop. Of
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Finish
In algebra, certain easy steps are left out,
because they are understood.
Start
Statements
Reasons
Given
Generally in algebra the reflexive steps are
invisible or left out for speed and/or
convenience.
4 4
Reflexive Prop.
3x 4( 7 x )
Mult. Prop. Of
Distr. Prop. Of
3x 28 4x
4x 4x
Reflexive Prop.
Prop. Of
Eventually, we will do the same. But not just yet!
7x 28
7 7
Reflexive Prop.
x 4
Div. Prop. Of
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Geometric Proof 1
If AB CD, then AC BD.
Given AB CD Prove AC BD
?
First step is to label the diagram.
g
g
A
D
C
B
?
Statements
Reasons
Labeling means marking and giving the reasons
next to the markings.
Given
AB CD
Reflexive Prop.
BC BC
Prop. Of
ABBC BCCD
Seg. Addition Post.
ABBC AC
Start with given and then add steps to reach the
conclusion.
BCCD BD
Seg. Post.
AC BD
Substitution
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Geometric Proof 2
If AB BE and DB CB, then AC DE.
g
g
A
E
Given AB BE DB CB Prove AC
BD
?
?
B
g
C
D
g
1st step is to label the diagram.
Statements
Reasons
Given
Labeling means marking and giving the reasons
next to the markings.
AB BE
Given
BC DB
Prop. Of
ABBC DBBE
Seg. Addition Post.
ABBC AC
Start with given and then add steps to reach the
conclusion.
DBBE DE
Seg. Post.
AC DE
Substitution
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Summary
1 The properties of algebra are used as reasons
or justifications of steps in proofs.
2 Four of the properties are associated with
arithmetic operations in equations
Euclid said it simply as Equals when
by equals are equal.
Added Subtracted Multiplied Divided
Addition Subtraction Multiplication Division
Each one is known as the
property of equality.
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Summary
3 The distributive property involves
parentheses.
a( b c ) ab ac
Multiplication is distributed to each item
inside the parentheses.
4 Proofs are a process of linking statement
together from the hypotheses to the conclusion.
It will take over a month to get comfortable with
the process of writing proofs.
Relax. Be patient. (hard to do) It WILL come.
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Cest fini.
Good day and good luck.