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Geometry

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

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Geometry

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

  2. 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

  3. Reflexive • a = a • Symmetric • If a = b, then b = a • Transitive • If a = b and b = c, then a = c

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

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

  6. Prove that if –5(x+4) = 70, then x = -18Write a justification for each step. -5(x + 4) = 70 Given Proof: 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

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

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

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

  10. 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.

  11. Given: Prove: Given Multiplication prop Sub Addition prop Sub Symmetric

  12. Write a two-column proof to verify.

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

  14. Properties

  15. Write a two column proof to verify the conjecture.

  16. Given: 1. 1. Given • Definition of • Congruent angles 2. Prove: 3. 3. Transitive x = 6 6x + 7 = 8x - 5 4. Sub 4. 5. 5. Subt. Prop. 6x + 7 – 8x =8x – 5 – 8x -2x + 7 = -5 6. 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

  17. 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. Sub 10. 6 = x 11. Symmetric 11. x = 6

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

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