Algebraic proof Chapter 2 Section 6

Properties of Equality 2 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 a * c = b * c Division Property If a = b and c ≠ 0, then a/c = b/c Reflexive Property a = a Symmetric Property If a = b, then b = a Transitive Property If a = b and b = c, then a = c Substitution Property If a = b, then b can replace a in any expression Distributive Property a(b + c) = ab + ac

Solve and Write a Justification for Each Step 3 Solve 6x + 2(x - 1) = 30 and write a justification for each step: 6x + 2(x - 1) = 30 6x + 2x - 2 = 30 8x - 2 = 30 8x = 32 8x = 32 8 8 x = 4 Given or original equation Distributive Property Combine Like Terms Addition property Division property of = Simplify

This leads to “Proof Writing”

How to Set Up a 2-column Proof Given: Prove: Statement Reason Diagram

Write a geometric proof Give a reason for each step Multiplication Distributive Addition

Write a 2-column proof Distributive Subtraction Division

Properties of Congruence Reflexive Property AB ≅ AB ∠ A ≅ ∠ A Symmetric Property If AB ≅ CD, then CD ≅ AB If ∠ A ≅ ∠ B, then ∠ B ≅ ∠ A Transitive Property If AB ≅ CD and CD ≅ EF, then AB ≅ EF If ∠ A ≅ ∠ B and ∠ B ≅ ∠ C,then ∠ A ≅ ∠ C

Using Properties of Equality and Congruence State the property that justifies each statement 1.Reflexive 2.Division 3.Multiplication 4.subtraction 5.Substitution 6.Addition 7.Transitive 1. 2. 3. 4. 5. 6. 7.

Write a 2-column proof If DF EG, then x = 10 ≅ D E Statements Reasons 11 1. DF = EG 2. 11 = 2x - 9 3. 20 = 2x 4. 10 = x 1. Given 2. Substitution 3. Addition 4. Division G F

Write a 2 –column proof 1. Given 1. <Y = <Z 2. Substitution If <Y <Z, then x = 100 ≅ Statements Reasons 1. <Y = <Z 2. x + 10 = 2x - 90 3. 10 = x - 90 4. 100 = x 1. Given 2. Substitution 3. Subtraction 4. Addition X +10 y Z 2x- 90

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