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Warm Up

Warm Up. Warm Up Answers. 2.5 Algebraic Proof. Monty Python’s Crazy Logic (click on the image to view video). 2.5 Algebraic Proof. Objectives : Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence.

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Warm Up

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  1. WarmUp

  2. Warm Up Answers

  3. 2.5 Algebraic Proof Monty Python’s Crazy Logic (click on the image to view video)

  4. 2.5 Algebraic Proof Objectives: Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

  5. Section 2-5: Reasoning in AlgebraStandard: apply reflective, transitive, or symmetric properties of equality or congruence • Objectives: • Connect reasoning in algebra and geometry • Justify steps in deductive reasoning • In geometry • postulates, definitions, & properties are accepted as true • (refer to page 842 for a complete list of postulates) • you use deductive reasoning to prove other statements • We will look at some basic properties used to justify statements….. • ….. which leads to writing proofs.

  6. Properties of Equality Page 113 Addition Property of Equality If a = b, then a + c = b + c Add same amount to both sides of an equation. Subtraction Property of Equality If a = b, then a - c = b - c Subtract same amount to both sides of an equation. Multiplication Property of Equality If a = b, then a ∙ c = b ∙ c Multiply both sides of an equation by the same amount. Division Property of Equality If a = b and c  0, then Divide both sides of an equation by the same amount.

  7. Properties of Equality (cont) Reflective Property of Equality a = a Ex: 5 = 5 Symmetric Property of Equality If a = b, then b = a Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same. Transitive Property of Equality If a = b and b = c, then a = c. EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2. Substitution Property of Equality If a = b , then b can replace a in any expression. Ex: a = 3; If a = b, then 3 = 3. Distributive Property a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9

  8. Remember! The Distributive Property states that a(b + c) = ab + ac. 2.5 Properties of EqualityTable on page #113

  9. Properties of Congruence The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements. Reflective Property of Congruence AB  AB A A Symmetric Property of Congruence If AB  CD, then CD AB. If A B, then B  A Transitive Property of Congruence If AB  CD and AB  EF, then CD EF. If A  B and B  C, then A  C.

  10. 2.5 Properties of CongruenceTable on page #114

  11. Helpful Hint • AB • AB represents the length AB, so you can think of AB as a variable representing a number. What’s the Difference between equality and congruence?

  12. Remember! Numbers are equal (=) and figures are congruent (). Congruence Equality Measurements (numbers)can be equal to each other. Geometric objects (figures / drawings) can be congruent to each other. Statements use symbol Statements use = symbol

  13. 2.5 Application Write a justification for each step. NO = NM + MO Segment Addition Post. 4x – 4 = 2x + (3x – 9)Substitution Property of Equality 4x – 4 = 5x – 9 Simplify. –4 = x – 9 Subtraction Property of Equality 5 = xAddition Property of Equality

  14. Given - facts you are given to use. STARTING POINT The basic format of a two column proof: Page 115 Prove – conclusion you need to reach. ENDING POINT

  15. Proof Example: Problem 3 page 116 This is how you plan to get from the given to the prove. This is given This is what you are asked to prove Reasons

  16. Application PROVE: y = 6 GIVEN: • Segment addition postulate • Substitution • Combine like terms • Addition Property (add 9 to both sides) • Division property (divide both sides by 5)

  17. Using Properties to Justify Steps in Solving Equations Algebra: Prove x = 43 and justify each step. Given: m AOC= 139 Prove : x = 43 Statement Reasons Angle Addition Postulate M AOB+ mBOC= mAOC x+ 2x + 10 = 139 Substitution Property Simplify or combine like terms 3x + 10 = 139 3x = 129 Subtraction Property of Equality x = 43 Division Property of Equality

  18. Using Properties to Justify Steps in Solving Equations Prove x = 20 and justify each step. Given: LM bisects KLN Prove: x = 20 Statement Reasons Def of Angle Bisector Substitution Property Subtraction Property of Equality Division Property of Equality

  19. Using Properties to Justify Steps in Solving Equations Now you try Solve for y and justify each step Given: AC = 21 Prove : y = 6 Statement Reasons Segment Addition Postulate AB +BC=AC 2y + 3y - 9 = 21 Substitution Property Simplify 5y – 9 = 21 5y = 30 Addition Property of Equality y = 6 Division Property of Equality Find AB and BC by substituting y = 6 into the expressions.

  20. Using Properties of Equality and Congruence Name the property of congruence or equality the justifies each statement. a. K  K Reflective Property of Congruence b. If 2x – 8 = 10, then 2x = 18 Addition Property of Equality c. If RS  TW and TW  PQ, then RS  PQ. Transitive Property of Congruence d. If m A = m B, then m B = m A Symmetric Property of Equality

  21. Use what you know about transitive properties to complete the following: The Transitive Property of Falling Dominoes: If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall. C

  22. HOMEWORK COMPLETE 2-5 PACKET DUE THURSDAY NOV 1

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