Download 2.6 Properties of Equality and Congruence

2.5 and 2.6 Properties of Equality
and Congruence
Objective:

You will use deductive reasoning to:
–
–
Write proofs using geometric theorem
To use algebraic properties in logical arguments.
Algebraic Properties

Substitution Property: If a = b, then a can be
substituted for b in an equation or
expression.

Distributive Property: a(b + c) = ab + ac,
where a, b, and c are real numbers.
Algebraic Properties

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 ac = bc.

Division Property:
If a = b and c = 0, then a/c = b/c.
Example 1:
Write a two-column proof to solve the
equation.
3x + 2 = 8
Statements
1.
2.
3.
4.
5.
3x + 2 = 8
3x + 2 – 2 = 8 – 2
3x = 6
3x ÷ 3 = 6 ÷ 3
x=2
Reasons
1. Given
2. Subtraction Prop
3. Simplify
4. Division Prop
5. Simplify
Example 2:
Write a two-column proof to solve the
equation.
Statements
Reasons
Given
6.
4x + 9 = 16 – 3x
4x + 9 + 3x = 16 – 3x + 3x
7x + 9 = 16
7x + 9 – 9 = 16 – 9
7x = 7
7x ÷ 7 = 7 ÷ 7
7.
x=1
Simplify
1.
2.
3.
4.
5.
Addition Prop
Simplify
Subtraction Prop
Simplify
Division Prop
Example 3:
Write a two-column proof to solve the
equation.
2(-x – 5) = 12
Statements
1.
2.
3.
4.
5.
6.
2(-x – 5) = 12
-2x – 10 = 12
-2x – 10 + 10 = 12 + 10
-2x = 22
-2x ÷ -2 = 22 ÷ -2
Prop
x = -11
Reasons
Given
Distributive Prop
Addition Prop
Simplify
Division
Simplify
Reflexive Property

Equality
AB  AB
mA  mA

Congruence
AB  AB
A  A
Symmetric Property

Equality:
If AB  CD, then CD  AB
If mA  mB, then mB  mA

Congruence:
If AB  CD, then CD  AB
If A  B, then B  A
Transitive Property

Equality:
If AB  CD & CD  EF , then AB  EF
If mA  mB & mB  mC , then mA  mC

Congruence:
If AB  CD & CD  EF , then AB  EF
If A  B & B  C , then A  C
Properties of Equality

Addition Property: adding a number to each side of an
equation

Subtraction Property: subtracting a number from each
side of an equation
Properties of Equality

Multiplication Property: multiplying by a
number on each side of an equation

Division Property: dividing by a number on
each side of an equation

Substitution Property: substituting a number
for a variable in an equation to produce an
equivalent equation
Definitions
Theorem:
A true statement that follows as a result of other
true statements.
Two-column proof:
Most commonly used. Has numbered
statements and reasons that show the logical
order of an argument.
Use the diagram and the given information to
complete the missing steps and reasons in the proof.
GIVEN: LK = 5, JK = 5, JK ≅ JL
PROVE: LK ≅ JL
K
Statements:
Reasons:
J
1.
2.
3.
4.
5.
6.
_______________
_______________
LK = JK
LK ≅ JK
JK ≅ JL
________________
1.
2.
3.
4.
5.
6.
Given
Given
Transitive Property
_______________
Given
Transitive Property
L