Download 2.5 Reasoning with properties from Algebra

2.5 Reasoning with properties
from Algebra
GEOMETRY
Goal 1: Using Properties from
Algebra – Properties of
Equality
In all of the following properties –
Let a, b, and c be real numbers
Properties of Equality
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 ca = cb
Division property:
If a = b, then a  b for c  0
c
c
Addition Property
This is the property that allows you to
add the same number to both sides of
an equation.
STATEMENT
x=5
REASON
given
3+x=8
Addition property
of equality
Subtraction Property
This is the property that allows you to
subtract the same number to both
sides of an equation.
STATEMENT
x=5
REASON
given
X-2=3
Subtraction
property of equality
Multiplication Property
This is the property that allows you to
multiply the same number to both sides
of an equation.
STATEMENT
x=5
REASON
given
3x = 15
Multiplication
property of equality
Division Property
This is the property that allows you to
divide the same number to both sides
of an equation.
STATEMENT
x=5
x 5

3 3
REASON
given
Division property of
equality
More Properties of Equality
Reflexive Property:
a = a.
Symmetric Property:
If a = b, then b = a.
Transitive Property:
If a = b, and b = c, then a = c.
Reflexive Property: a = a
I know what you are thinking, duh this doesn’t seem
too difficult to grasp. Just remember this one, when
we begin to prove that triangles are congruent.
STATEMENT
x=x
REASON
Reflexive property
of equality
Symmetric Property:
a = b so b = a
I know another duh property. Just remember when
you get an answer that is a little different
than the one you are use to getting. (Do we like
To always have x or y on the left side of the equal sign?)
For example:
2 – y = 10
Transitive Property
This one is many times confused with substitution property
of equality.
Remember transitive is like “transit” which means to move.
Think of there being 3 bus stops: a, b, and c. If you move
from a to b, then from b to c, it would have been the same
as moving from a to c directly.
STATEMENT
REASON
mA =43o
given
mB =43o
given
mA = mB
Transitive property of equality
Substitution Property of
Equality
If a = b, then a may be substituted for b in any equation
or expression.
You have used this many times in algebra.
STATEMENT
x=5
3+x=y
3+5=y
REASON
given
given
substitution
property of equality
Distributive Property
a(b+c) = ab + ac
ab + ac = a(b+c)
STATEMENT
mA + mA
=90o
2mA =90o
REASON
given
Distributive
property
Properties of Congruence
Reflexive
object A  object A
Symmetric
If object A  object B, then object B  object A
Transitive
If object A  object B and object B  object C,
then object A  object C