1. Section 2-4: Reasoning in Algebra TPI 32A: apply reflective, transitive, or symmetric prooperties of equality or congruence
Objectives:
Connect reasoning in algebra and geometry
Justify steps in deductive reasoning
In geometry
postulates, definitions, & properties are accepted as true
you use deductive reasoning to prove other statements
We will look at some basic properties used to justify statements…..
….. which leads to writing proofs.

2. Properties of Equality
Addition Property of Equality
If a = b, then a + c = b + c
Add same amount to both sides of an equation.
Subtraction Property of Equality
If a = b, then a - c = b - c
Subtract same amount to both sides of an equation.
Multiplication Property of Equality
If a = b, then a ∙ c = b ∙ c
Multiply both sides of an equation by the same amount.
Division Property of Equality
If a = b and c  0, then
Divide both sides of an equation by the same amount.

3. Properties of Equality (cont)
Reflective Property of Equality
a = a Ex: 5 = 5
Symmetric Property of Equality
If a = b, then b = a Ex: 3 = and = 3 are the same.
Transitive Property of Equality
If a = b and b = c, then a = c.
EX: If = 7 and = 7, then =
Substitution Property of Equality
If a = b , then b can replace a in any expression.
Ex: a = 3; If a = b, then 3 = 3.
Distributive Property
a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9

4. Using Properties to Justify Steps in Solving Equations
Algebra Solve for x and justify each step.
Given: m AOC = 139
m AOC = 139
Given
m AOB + m BOC = m AOC
Angle Addition Postulate
x x = 139
Substitution Property
Simplify
3x + 10 = 139
3x = 129
Subtraction Property of Equality
x = 43
Division Property of Equality

5. Using Properties to Justify Steps in Solving Equations
Solve for x and justify each step.
Given: LM bisects KLN
LM bisects KLN
Given
MLN = KLM
4x = 2x + 40
2x = 40
x = 20
Def of Angle Bisector
Substitution Property
Subtraction Property of Equality
Division Property of Equality

6. Using Properties to Justify Steps in Solving Equations
Solve for y and justify each step
Given: AC = 21
AC = 21
Given
AB + BC = AC
Segment Addition Postulate
2y + 3y - 9 = 21
Substitution Property
Simplify
5y – 9 = 21
5y = 30
Addition Property of Equality
y = 6
Division Property of Equality
Find AB and BC by substituting y = 6 into the expressions.

7. Properties of Congruence
The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements.
Reflective Property of Congruence
AB  AB
A  A
Symmetric Property of Congruence
If AB  CD, then CD  AB.
If A  B, then B  A
Transitive Property of Congruence
If AB  CD and AB  EF, then CD  EF.
If A  B and B  C, then A  C.

8. Using Properties of Equality and Congruence
Name the property of congruence or equality the justifies each statement.
a. K  K
Reflective Property of 
b. If 2x – 8 = 10, then 2x = 18
Addition Property of Equality
c. If RS  TW and TW  PQ,
then RS  PQ.
Transitive Property of 
d. If m A = mB, then
m B = mA
Symmetric Property of Equality

9. Use what you know about transitive properties to complete the following:
The Transitive Property of Falling Dominoes:
If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall. C