1. 2.5 Reason Using Properties from Algebra Objective: To use algebraic properties in logical arguments.

2. 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.

3. 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.

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

5. Example 2: Write a two-column proof to solve the equation. StatementsReasons 1.4x + 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

6. Example 3: Write a two-column proof to solve the equation. StatementsReasons 1.2(-x – 5) = 12Given 2.-2x – 10 = 12Distributive Prop 3.-2x – 10 + 10 = 12 + 10Addition Prop 4.-2x = 22Simplify 5.-2x ÷ -2 = 22 ÷ -2 Division Prop 6.x = -11Simplify 2(-x – 5) = 12

7. Algebraic Properties Reflexive Property: For any real number a, a = a For 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 = a For any segments AB and CD, if AB = CD, then CD = AB For any angles A and B, if m<A = m<B, then m<B = m<A

8. 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

9. 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 StatementReason AB + BC = CD + BC Addition Property of Equality AC = BD Substitution Property of Equality

10. 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 StatementReason m 1 + m 2 = m DBC Angle Addition Postulate m EBA = m DBC Transitive Property of Equality

11. 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

12. 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