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Monty Python’s Crazy Logic (click on the image to view video) 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. Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

Section 2-5: Reasoning in Algebra Standard: 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.

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.

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

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

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.

2.5 Properties of Congruence Table on page #114

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

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

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 = x Addition Property of Equality

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

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

Application Statement Reason AB + BC = AC 2y + 3y – 9 = 21 5y – 9 = 21 PROVE: y = 6 GIVEN: Statement Reason AB + BC = AC 2y + 3y – 9 = 21 5y – 9 = 21 5y = 30 y = 6 Segment addition postulate Substitution Combine like terms Addition Property (add 9 to both sides) Division property (divide both sides by 5)

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 m AOC = 139 Given M AOB + m BOC = m AOC Angle Addition Postulate 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

Using Properties to Justify Steps in Solving Equations Prove x = 20 and justify each step. Given: LM bisects KLN Prove: x = 20 Statement Reasons LM bisects KLN Given MLN = KLM 4x = 2x + 40 2x = 40 x = 20 Def of Angle Bisector Substitution Property Subtraction Property of Equality Division Property of Equality

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 AC = 21 Given AB + BC = AC Segment Addition Postulate 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.

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

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

HOMEWORK COMPLETE 2-5 PACKET DUE THURSDAY NOV 1