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Geometry CP
Outline Sec. 2-5
Mrs. Sweet
Name:
Date:
Reasoning in Algebra and Geometry
There are 3 types of properties in mathematics:
Assumed properties or
Defining properties or
Deduced properties or
Postulates describe
terms and act as starting points for logical deductions.
These properties hold for all real numbers a, b, and c.
Properties of
Equality
Definition
Reflexive Property
a=a
Symmetric Property
If a = b, then b = a
Transitive Property
If a = b and b= c, then a = c
Equality Properties
Definition
Addition Property of Equality
If a = b, then a + c = b + c
Multiplication Property of
Equality
If a = b, then ac = bc
Substitution Property of Equality
If a = b then b can replace a in
any equation
Example
Ex. 1: Solve the following. State the properties of equality used to solve the equation.
2x – 4 = 12
Example
Postulates of Inequality
Definition
Transitive Property of Inequality
If a < b and b < c, then a < c
Addition Property of Inequality
If a < b , then a + c < b + c
Example
If a < b and c > 0, then ac < bc
Multiplication Properties of
Inequality
If a < b and c < 0 then ac > bc
Ex.2: Name the properties of inequality and
operations that are used in solving the following
inequality.
1
180   m  20
4
Ex. 3: Knowing that
Don’t forget to flip
the inequality symbol
when multiplying or
dividing by a
negative!
mX  mY what can you conclude about mX  mZ and mY  mZ ?
Remember these?
Commutative Property of Addition
Commutative Property of
Multiplication
Associative Property of Addition
Associative Property of
Multiplication
Distributive Property
Ex. 4 What is the value of x? Justify each step.
Given: AOM and MOC are supplementary
Some properties of equality have
properties of congruence.
Using properties of Equality and Congruence:
What is the name of the property of equality or congruence that justifies going from the first statement to the
second statement:
EX. 5.
EX. 6.
3x  y
y
x
3
ST  ST
ST  ST
A
why a conjecture is true.
EX. 7. 6x – 3
3(2x – 1)
is a convincing argument that uses deductive reasoning. A proof logically shows
Two Column Proof:
Lists each
on the right and each
Developing Proof Fill in the missing statements or reasons for the following two-column proof.
Given: AB is the bisector of CAD.
Prove: x = 9
Statements
Reasons
1) AB is the bisector of CAD.
1) Given
2) CAB  BAD
2) ?
3) mCAB  mBAD
3)  angles have equal measures.
4) 7x + 2 = 5(x + 4)
4) ?
5) ?
5) 7x + 2 = 5x + 20
6) ?
7) ?
?
6)
7) ?
on the left.