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.
“Indubitably.”
“The proof is in the pudding.”
Le pompt de pompt le solve de crime!"
Deductive Reasoning
Je solve le crime. Pompt de pompt pompt."
Properties of Equality
Addition property of equality
If a = b and c = d, then a + c = b + d.
restated
If a = b and
c = d, then
a + c = b + d.
Example
a=b
3=3
a+3=b+3
Euclid referred to this property as…
“Equals when added to equals are equal.”
Properties of Equality
Subtraction property of equality
If a = b and c = d, then a - c = b - d.
restated
If a = b and
c = d, then
a - c = b - d.
Example
a=b
3=3
a-3=b-3
Euclid referred to this property as…
“Equals when subtracted from equals are equal.”
Properties of Equality
Multiplication property of equality
If a = b and c = d, then ac = bd.
restated
If a = b and
c = d, then
ac = bd.
Example
a=b
3=3
3a = 3b
Euclid referred to this property as…
“Equals when multiplied by equals are equal.”
Properties of Equality
Division property of equality
a
b
If a = b and c  0 , then 
c c
restated
If a = b and c  0
c = c, then
a b

c c
Example
a=b
3=3
a b

3 3
Euclid referred to this property as…
“Equals when divided by equals are equal.”
Properties of Equality
Division property of equality
a b
If a = b and c  0, then 
c c
restated
If a = b and c  0
c = c, then Why must c not equal zero?
a b

c c
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.”
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.
Properties of Equality
Symmetric property of equality
If a = b, then b = a.
This is really obvious. Nevertheless, it needs a name.
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.
Properties of Congruence
Reflexive property of congruence
DE  DE
and
 D  D
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.
Didn’t we say this before? YES
Properties of Congruence
Symmetric property of Congurence
If AB  DE , then DE  AB .
If  A   B , then  B   A .
This is really obvious. Nevertheless, it needs a name.
Symmetric is the same for equality and congruence.
Properties of Congruence
Transitive property of Congruence
If  A   B and  B   C, then  A   C.
If AB  BC and BC  CD, then AB  CD.
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.
Distributive Property
a( b + c) = ab +ac
Implied multiplication
Recognition of Properties 1
If AB = CD , then
AB  CD.
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
AB  CD.
then AB = CD.
Definition of congruent segments.
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
Why?
Addition property of equality.
I added 3 to both sides. Remember Euclid?
“Equals when added to equals are equal.”
Recognition of Properties 3
If B is on line AC and AB = BC ,
then b is the midpoint of AC
Definition of midpoint.
If A = B , then A + 3 = B + 3.
+3 +3
Addition property of equality.
If AB  BC , then BC  AB .
switch sides
 A  A
Mirror image
Symmetric property of equality.
Reflexive property of equality.
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
a b

c c
Division property of equality.
Recognition of Properties 5
If
7a
7
7
 6 , then
a = 42.
Multiplication property of equality.
If __
8x = 48,
___ then x = 6. Division property of equality.
8
8
If 2y – 7 = 11, then 2y = 18.
+7
If
+7
Addition property of equality.
 X  Y then  Y   X
switch sides
Symmetric property of equality.
Recognition of Properties 6
If AB = 30 and A = 5 , then 5B = 30.
Substitution property of equality.
If B is the midpoint of
AC
, then AB = BC.
Definition of midpoint.
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
7x + 14 = 35
14 = 14
7x = 21
7=7
x=3
Solving a first degree equation
with 1 variable.
Proofs
The major difference between
equations and geometric proofs is
in the form.
If 7( x + 2 ) = 35, then x =3.
Statements
7( x + 2 ) = 35
7x + 14 = 35
14 = 14
7x = 21
7=7
x=3
Written as a conditional.
Reasons
Given Information
Distributive Property
Reflexive Property
Subtraction Prop. Of Equality
Reflexive Property
Division Prop. Of Equality
The only difference is that the reasons/justification
for each step must be written in geometry.
3
If x  7  x , then x  4.
4 Start
Finish
Statements
3
x7  x
4
4=4
Reasons
Given
Note this is a
lot of writing.
You will need
To abbreviate
Reflexive Property
3x = 4( 7 – x )
Multiplication Prop. Of Equality
3x = 28 – 4x
Distributive Property
4x = 4x
Reflexive Property
Addition Prop. Of Equality
Reflexive Property
Division Prop. Of Equality
7x = 28
7=7
x=4
3
If x  7  x , then x  4.
4 Start
Finish
Statements
3
x7  x
4
4=4
Reasons
Given
Reflexive Prop.
3x = 4( 7 – x )
Mult. Prop. Of =
3x = 28 – 4x
Distr. Prop. Of =
4x = 4x
Reflexive Prop.
+ Prop. Of =
Reflexive Prop.
Div. Prop. Of =
7x = 28
7=7
x=4
This is a lot
less writing.
3
If x  7  x , then x  4.
4 Start
Finish In algebra, certain
Statements
3
x7  x
4
4=4
Reasons
Given
Reflexive Prop.
3x = 4( 7 – x )
Mult. Prop. Of =
3x = 28 – 4x
Distr. Prop. Of =
4x = 4x
Reflexive Prop.
+ Prop. Of =
Reflexive Prop.
Div. Prop. Of =
7x = 28
7=7
x=4
easy steps are left
out, because they
are understood.
Generally in algebra
the reflexive steps
are invisible or left
out for speed and/or
convenience.
Eventually, we
will do the same.
But not just yet!
Geometric Proof 1
If AB = CD, then AC = BD.
Given: AB = CD
Prove: AC = BD
?
A
g B
Statements
AB = CD
BC = BC
AB+BC = BC+CD
AB+BC = AC
BC+CD = BD
AC = BD
g D
C
?
Reasons
Given
Reflexive Prop.
First step is to
label the diagram.
Labeling means
marking and
giving the
reasons next to
the markings.
+ Prop. Of =
Seg. Addition Post. Start with given
and then add steps
Seg. + Post.
to reach the
Substitution
conclusion.
Geometric Proof 2
g g
A
E
?
D
g
B
Statements
AB = BE
BC = DB
If AB = BE and DB = CB,
then AC = DE.
?
g
C
Reasons
Given
Given
AB+BC = DB+BE
AB+BC = AC
+ Prop. Of =
Seg. Addition Post.
DB+BE = DE
Seg. + Post.
AC = DE
Substitution
Given: AB = BE
DB = CB
Prove: AC = BD
1st step is to label
the diagram.
Labeling means
marking and
giving the
reasons next to
the markings.
Start with given
and then add steps
to reach the
conclusion.
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:
Added
Equals when Subtracted by equals are equal.
Multiplied
Divided
Addition
Each one is known as the Subtraction property of equality.
Multiplication
Division
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.
C’est fini.
Good day and good luck.