2.5 Reason Using Properties from Algebra

Objective: To use algebraic properties in logical arguments.

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.

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.

Example 1: Write a two-column proof to solve the equation.

Statements Reasons

1. 3x + 2 = 8

2. 3x + 2 – 2 = 8 – 2

3. 3x = 6

4. 3x ÷ 3 = 6 ÷ 3

5. x = 2

3x + 2 = 8

Given

Subtraction Prop

Simplify

Division Prop

Simplify

Example 2: Write a two-column proof to solve the equation.

Statements Reasons1. 4x + 9 = 16 – 3x

2. 4x + 9 + 3x = 16 – 3x + 3x

3. 7x + 9 = 16

4. 7x + 9 – 9 = 16 – 9

5. 7x = 7

6. 7x ÷ 7 = 7 ÷ 7

7. x = 1

Given

Addition Prop

Simplify

Subtraction Prop

Simplify

Division Prop

Simplify

Example 3: Write a two-column proof to solve the equation.

Statements Reasons

1. 2(-x – 5) = 12 Given

2. -2x – 10 = 12 Distributive Prop

3. -2x – 10 + 10 = 12 + 10 Addition Prop

4. -2x = 22 Simplify

5. -2x ÷ -2 = 22 ÷ -2 Division Prop

6. x = -11 Simplify

2(-x – 5) = 12

Algebraic Properties• Reflexive Property:

For any real number a, a = aFor any segment AB, AB = AB For any angle A, m<A = m<A

• Symmetric Property:For any real numbers a and b, if a = b, then b = aFor any segments AB and CD, if AB = CD, then CD = ABFor any angles A and B, if m<A = m<B, then m<B = m<A

Algebraic Properties (cont)Transitive Property:

For any real numbers a, b and c, if a = b and b = c, then a = c.

For any segments AB, CD, and EF, if AB = CD and CD = EF, then AB = EF.

For any angles A, B and C, if m<A = m<B, and m<B = m<C then m<A = m<C

In the diagram, AB = CD. Show that AC = BD.

AB = CD Given

AC = AB + BC Segment Addition Postulate

BD = BC + CD Segment Addition Postulate

Example 4

Statement Reason

AB + BC = CD + BC Addition Property of Equality

AC = BD Substitution Property of Equality

You are designing a logo to sell daffodils. Use the information given. Determine whether m EBA = m DBC.

m 1 = m 3 Given

m EBA = m 3+ m 2 Angle Addition Postulate

m EBA = m 1+ m 2 Substitution Property of Equality

Example 5

Statement Reason

m 1 + m 2 = m DBC Angle Addition Postulate

m EBA = m DBC Transitive Property of Equality

Example 5:

Name the property of equality the statement illustrates.

Symmetric Property of Equality

ANSWER

b). If JK = KL and KL = 12, then JK = 12.

ANSWER

Transitive Property of Equality

a). If m 6 = m 7, then m 7 = m 6.

Example 6

Example 5 cont’d:

c). m W = m W

ANSWER

Reflexive Property of Equality

d). If L = M and M = 6, then L = 6

ANSWER

Transitive Property of Equality