1. Reasoning in Algebra and Geometry
Section 2.5

2. How to justify Algebra and Geometry
We use properties of equality, properties of congruence, and postulates to justify each step that is taken.
Proof
A convincing argument that uses deductive reasoning.
Two ways to write proofs
Two-column proof
Lists statements on the left and reasons for each statement on the right.
Paragraph proof
The statements and reasons are written in sentences.

3. Properties of Equality
Addition Property
Subtraction Property
Multiplication Property
Division Property
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
If a=b, then a + c = b + c
If a=b, then a – c = b – c
If a=b, then a * c = b * c
If a=b and c ≠ 0, then a/c = b/c
a = a
If a = b, then b = a
If a = b and b = c, then a = c
If a = b, then b can replace a in any expression.

4. The Distributive Property
Use multiplication to distribute a to each term within the parentheses.
a(b + c) = ab + ac
a(b – c) = ab – ac

5. Properties of Congruence
Reflexive Property Symmetric Property Transitive Property
If , then
If and , then
If and , then
If and , then

6. Example 1 What is the value of x? Justify each step.
Angle AOM and angle MOC = 180 Angles form a linear pair
Definition of supplementary lines
(2x + 30) + x = Substitution Property
3x + 30 = Addition Property
Subtraction Property
3x = Division Property
x = Substitution Property x
(2x + 30) A M O C

7. Example 2
What is the name of the property of equality or congruence that justifies going from first to second statement?
3(x + 5) = 9
3x + 15 = 9
¼ x = 7
x = 28

8. Example 3 Two-Column Proof
Given: 5x + 1 = 21
Prove: x = 4
Statements Reasons
5x + 1 = Given
5x = Subtraction Prop of Equality
X = Division Prop. of Equality

9. Assignment pg
#5-19
Show work