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2.5 Reason Using Properties from Algebra

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.

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2.5 Reason Using Properties from Algebra

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  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. 3x + 2 = 8 StatementsReasons • 3x + 2 = 8 • 3x + 2 – 2 = 8 – 2 • 3x = 6 • 3x ÷ 3 = 6 ÷ 3 • x = 2 Given Subtraction Prop Simplify Division Prop Simplify

  5. Example 2: Write a two-column proof to solve the equation. StatementsReasons • 4x + 9 = 16 – 3x • 4x + 9 + 3x = 16 – 3x + 3x • 7x + 9 = 16 • 7x + 9 – 9 = 16 – 9 • 7x = 7 • 7x ÷ 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. 2(-x – 5) = 12 StatementsReasons • 2(-x – 5) = 12 Given • -2x – 10 = 12 Distributive Prop • -2x – 10 + 10 = 12 + 10 Addition Prop • -2x = 22 Simplify • -2x ÷ -2 = 22 ÷ -2 Division Prop • x = -11 Simplify

  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. Example 4 In the diagram, AB = CD. Show that AC = BD. Statement Reason AB = CD Given AC = AB + BC Segment Addition Postulate BD = BC + CD Segment Addition Postulate AB + BC = CD + BC Addition Property of Equality AC = BD Substitution Property of Equality

  10. Example 5 You are designing a logo to sell daffodils. Use the information given. Determine whether mEBA=mDBC. m 1 + m 2 = mDBC mEBA = mDBC m 1 = m 3 mEBA =m 3+ m 2 mEBA =m 1+ m 2 Statement Reason Given Angle Addition Postulate Substitution Property of Equality Angle Addition Postulate Transitive Property of Equality

  11. a). If m 6 = m 7, then m 7 = m 6. ANSWER Symmetric Property of Equality ANSWER Transitive Property of Equality Example 6 Example 5: Name the property of equality the statement illustrates. b). If JK = KLand KL = 12, then JK = 12.

  12. c). m W = m W ANSWER Reflexive Property of Equality ANSWER Transitive Property of Equality Example 5 cont’d: d). If L = M and M = 6, then L = 6

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