Lesson 2 – 6 Algebraic Proof Geometry Lesson 2 – 6 Algebraic Proof Objective: Use algebra to write two-column proofs. Use properties of equality to write geometric proofs.

Algebraic properties Addition Property of Equality If a = b, then a + c = b + c Subtraction Property of Equality If a = b, then a – c = b – c Multiplication Property of Equality If a = b, then a(c) = b(c) Division property If a = b, the a/c = b/c c cannot be 0

Reflexive Symmetric Transitive a = a If a = b, then b = a If a = b and b = c, then a = c

Distributive Property Substitution If a = b, then a may be replaced by b in any equation or expression. Distributive Property a(b + c) = ab + ac

Algebraic proof A proof that is made up of a series of algebraic statements.

Distributive property -5x - 20 = 70 +20 +20 Subtraction prop -5x = 90 Prove that if –5(x+4) = 70, then x = -18 Write a justification for each step. Proof: -5(x + 4) = 70 Given Distributive property -5x - 20 = 70 +20 +20 Subtraction prop -5x = 90 Substitution Note: Must rewrite When asked to show steps Division prop. Substitution x = -18

State the property that justifies each statement. If 4 + (-5) = -1, then x + 4 + (-5) = x – 1 If 5 = y, then y = 5. Addition property Symmetric property

Solve 2(5 – 3a) – 4(a + 7) = 92. Write a Justification for each step. Given 2(5 – 3a) – 4(a + 7) = 92 10 – 6a – 4a – 28 = 92 Distributive prop -18 – 10a = 92 Sub. Add. prop +18 +18 sub -10a = 110 Division prop sub a = -11

Two – Column Proof Statements and reasons organized in to columns.

Real world problem If the formula to convert a Fahrenheit temperature to a Celsius temperature is ,then the formula to convert a Celsius temperature to a Fahrenheit temperature is . Write a two-column proof to verify this conjecture.

Given: Prove: Given Sub Sub Symmetric Statements Reasons 1. 2. 3. 4. 5. 6. Given Multiplication prop Sub Addition prop Sub Symmetric

Write a two-column proof to verify.

Given: Prove: x = 3 Statements Reasons 1. Given: 1. 2. Add. Prop. 2. 3. Sub 3. 4. Mult. Prop. 4. 5. 5x + 1 = 16 5. Sub 6. 5x + 1 – 1 = 16 - 1 6. Subt. Prop. 7. 5x = 15 7. Sub 8. 8. Division Prop. 9. x = 3 9. Sub

Properties

Write a two column proof to verify the conjecture.

Given: Prove: Statements Reasons 1. 1. Given Definition of Congruent angles 2. Prove: 3. 3. Transitive x = 6 4. 6x + 7 = 8x - 5 4. Sub 5. 6x + 7 – 8x =8x – 5 – 8x 5. Subt. Prop. 6. -2x + 7 = -5 6. Sub 7. -2x + 7 – 7 = -5 - 7 7. Subt. Prop. 8. -2x = -12 8. Sub You do not need the Picture This is just so we can see it 9. 9. Division Prop. 10. x = 6 10. Sub

What if you had Solved the equation differently? Have to have Statements Reasons 1. 1. Given What if you had Solved the equation differently? Definition of Congruent angles 2. 3. 3. Transitive 4. 6x + 7 = 8x - 5 4. Sub 5. 6x+7 – 6x = 8x – 5 - 6x 5. Subt. Prop. 6. 7 = 2x - 5 6. Sub 7. 7 + 5 = 2x – 5 + 5 7. Add. Prop. 8. 12 = 2x 8. Sub 9. 9. Division Prop. Have to have this step! 10. 6 = x 10. Sub 11. x = 6 11. Symmetric

Homework Pg. 137 1 – 5 all, 10 – 18 E, 24