1. 2.5 Reasoning with properties from Algebra
GEOMETRY

2. Goal 1: Using Properties from Algebra – Properties of Equality
In all of the following properties –
Let a, b, and c be real numbers

3. 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 for c  0

4. Addition property of equality
This is the property that allows you to add the same number to both sides of an equation.
STATEMENT
REASON
x = 5
given
3 + x = 8
Addition property of equality

5. Subtraction property of equality
This is the property that allows you to subtract the same number to both sides of an equation.
STATEMENT
REASON
x = 5
given
X - 2 = 3
Subtraction property of equality

6. Multiplication Property
This is the property that allows you to multiply the same number to both sides of an equation.
STATEMENT
REASON
x = 5
given
3x = 15
Multiplication property of equality

7. Division property of equality
This is the property that allows you to divide the same number to both sides of an equation.
STATEMENT
REASON
x = 5
given
Division property of equality

8. 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.

9. 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
REASON
x = x
Reflexive property of equality

10. 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

11. Transitive property of equality
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
mA = mB
Transitive property of equality

12. 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
REASON
x = 5
3 + x = y
given
3 + 5 = y
substitution property of equality

13. Distributive Property
a(b+c) = ab + ac
ab + ac = a(b+c)
STATEMENT
REASON
mA + mA =90o
given
2mA =90o
Distributive property

14. 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