properties of algebra there are various properties from algebra that allow us to perform certain...

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.

Deductive Reasoning

“The proof is in the pudding.”“Indubitably.”

Je solve le crime. Pompt de pompt pompt."

Le pompt de pompt le solve de crime!"

Properties of Equality

If a = b and c = d, then a + c = b + d.

Addition property of equality

restated

If a = b and c = d, then a + c = b + d.

Example

a = b 3 = 3a + 3 = b + 3

Euclid referred to this property as…

“Equals when added to equals are equal.”


If a = b and c = d, then a - c = b - d.

Subtraction property of equality

restated

If a = b and c = d, then a - c = b - d.

Example

a = b 3 = 3a - 3 = b - 3


“Equals when subtracted from equals are equal.”


If a = b and c = d, then ac = bd.

Multiplication property of equality

restated

If a = b and c = d, then ac = bd.

Example

a = b 3 = 3 3a = 3b


“Equals when multiplied by equals are equal.”


If a = b and , then

Division property of equality

restated

If a = b and c = c, then

Example

a = b 3 = 3


“Equals when divided by equals are equal.”

a b

c c

3 3

a b

0c

a b

c c0c


If a = b and , then

Division property of equality

restated

If a = b and c = c, then


“Equals when divided by equals are equal.”

a b

c c

0c

0c a b

c c

Why must c not equal zero?

You are not allowed to divide by zero.

Numbers divided by zero are undefined.


a = a

Reflexive property of equality

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.


If a = b, then b = a.

Symmetric property of equality



If a = b and b = c, then a = c.

Transitive property of equality


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 CongruenceReflexive property of congruence


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.

DE DE and D D

Didn’t we say this before? YES

Properties of Congruence

If , then .

Symmetric property of Congurence


AB DE DE AB

If , then . A B B A

Symmetric is the same for equality and congruence.

Properties of CongruenceTransitive property of Congruence

This is the same as in equality.

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.

, .If A Band B C then A C

, .If AB BC and BC CD then AB CD

Distributive Property

a( b + c) = ab +ac

Implied multiplication

Recognition of Properties 1

If AB = CD , then .AB CDDefinition of congruent segments.

If then AB = CD..AB CD

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.

Definition of congruent segments.


If a = b and x = y , then a + x = b + y.

If a = b and x = y , then a - x = b - y.

If a = 7, then a + 3 = 10.

Addition property of equality.


Why? I added 3 to both sides. Remember Euclid?

“Equals when added to equals are equal.”

Subtraction property of equality.

+3 +3

-x -y

+x +y


If B is on line AC and AB = BC , then b is the midpoint of AC

If A = B , then A + 3 = B + 3.

, .If AB BC then BC AB

A A

+3 +3 Addition property of equality.

Definition of midpoint.

Symmetric property of equality. switch sides

Reflexive property of equality.Mirror image

11( 4x + 7) = 44x + 77


If a = b and b = c and c = 11, then a = 11.

If a = 11 , then a – 3 = 8.

If a = b and c = 12, thena b

c c

Distributive property.

Transitive property of equality.

-3 -3

c c

Subtraction property of equality.

Division property of equality.


If , then a = 42.67

a

If 8x = 48, then x = 6.

If 2y – 7 = 11, then 2y = 18.

If then X Y Y X

Multiplication property of equality.

Division property of equality.


Symmetric property of equality. switch sides

7 7

__ ___8 8

+7 +7


If B is the midpoint of , then AB = BC.

If AB = 30 and A = 5 , then 5B = 30.

AC

Substitution property of equality.

Definition of midpoint.

ProofsYou 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.

7( x + 2 ) = 35

7x + 14 = 35

14 = 14

7x = 21

7 = 7

x = 3

If 7( x + 2 ) = 35, then x =3.

Statements Reasons

Given InformationDistributive 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.

Written as a conditional.

3x = 4( 7 – x )

7 = 7

4x = 4x

4 = 4

x = 4

Statements Reasons

Given

Distributive Property

Reflexive Property

Multiplication Prop. Of Equality

Reflexive Property

Division Prop. Of Equality

37 , 4.

4If x x then x

Start Finish

37

4x x

3x = 28 – 4x

7x = 28 Addition Prop. Of Equality

Reflexive Property

Note this is a lot of writing.

You will needTo abbreviate

3x = 4( 7 – x )

7 = 7

4x = 4x

4 = 4

x = 4

Statements Reasons

Given

Distr. Prop. Of =

Reflexive Prop.

Mult. Prop. Of =

Reflexive Prop.

Div. Prop. Of =

37 , 4.

4If x x then x

Start Finish

37

4x x

3x = 28 – 4x

7x = 28 + Prop. Of =

Reflexive Prop.

This is a lot less writing.

3x = 4( 7 – x )

7 = 7

4x = 4x

4 = 4

x = 4

Statements Reasons

Given

Distr. Prop. Of =

Reflexive Prop.

Mult. Prop. Of =

Reflexive Prop.

Div. Prop. Of =

37 , 4.

4If x x then x

Start Finish

37

4x x

3x = 28 – 4x

7x = 28 + Prop. Of =

Reflexive Prop.

In algebra, certain easy steps are left out, because they are understood.

Eventually, wewill do the same.But not just yet!

Generally in algebra the reflexive steps are invisible or left out for speed and/or convenience.

Geometric Proof 1 If AB = CD, then AC = BD.

A DCB

Given: AB = CDProve: AC = BD

First step is to label the diagram.

?

?

g g

Statements Reasons

AB = CD

BC = BC

AB+BC = BC+CD

AB+BC = AC

BC+CD = BD

AC = BD

Given

Reflexive Prop.

+ Prop. Of =

Seg. Addition Post.

Seg. + Post.

Substitution

Labeling means marking and giving the reasons next to the markings.

Start with given and then add steps to reach the conclusion.

g

Geometric Proof 2 If AB = BE and DB = CB, then AC = DE.

A

D CB

Given: AB = BE DB = CBProve: AC = BD

1st step is to label the diagram.

??

g g

Statements Reasons

AB = BE

BC = DB

AB+BC = DB+BE

AB+BC = AC

DB+BE = DE

AC = DE

Given

+ Prop. Of =

Seg. Addition Post.

Seg. + Post.

Substitution

Labeling means marking and giving the reasons next to the markings.

Start with given and then add steps to reach the conclusion.

E

g

Given

Summary1 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.AddedSubtractedMultipliedDivided

Each one is known as the property of equality.

AdditionSubtractionMultiplicationDivision

Summary3 The distributive property involves parentheses.

Multiplication is distributed to each item inside the parentheses.

a( b + c ) = ab + ac

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.

properties of algebra there are various properties from algebra that allow us to perform certain...

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properties of equality

b euclid

symmetric property of

reflexive property of

division property of

addition property of

distributive property

properties of algebra