Geometry Lesson 2 6 Algebraic Proof Objective Use

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

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

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

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

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

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

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

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

Properties

Write a two column proof to verify the conjecture.

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

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

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