3 Monty Python’s Crazy Logic (click on the image to view video) 2.5 Algebraic ProofMonty Python’s Crazy Logic(click on the image to view video)
4 2.5 Algebraic ProofObjectives: 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 Algebra Standard: apply reflective, transitive, or symmetric properties of equality or congruenceObjectives:Connect reasoning in algebra and geometryJustify steps in deductive reasoningIn geometrypostulates, definitions, & properties are accepted as true(refer to page 842 for a complete list of postulates)you use deductive reasoning to prove other statementsWe will look at some basic properties used to justify statements…..….. which leads to writing proofs.
6 Properties of Equality Page 113Addition Property of EqualityIf a = b, then a + c = b + cAdd same amount to both sides of an equation.Subtraction Property of EqualityIf a = b, then a - c = b - cSubtract same amount to both sides of an equation.Multiplication Property of EqualityIf a = b, then a ∙ c = b ∙ cMultiply both sides of an equation by the same amount.Division Property of EqualityIf a = b and c 0, thenDivide both sides of an equation by the same amount.
7 Properties of Equality (cont) Reflective Property of Equalitya = a Ex: 5 = 5Symmetric Property of EqualityIf a = b, then b = a Ex: 3 = and = 3 are the same.Transitive Property of EqualityIf a = b and b = c, then a = c.EX: If = 7 and = 7, then =Substitution Property of EqualityIf a = b , then b can replace a in any expression.Ex: a = 3; If a = b, then 3 = 3.Distributive Propertya(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9
8 2.5 Properties of Equality Table on page #113 The Distributive Property states thata(b + c) = ab + ac.Remember!
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 CongruenceAB ABA ASymmetric Property of CongruenceIf AB CD, then CD AB.If A B, then B ATransitive Property of CongruenceIf AB CD and AB EF, then CD EF.If A B and B C, then A C.
10 2.5 Properties of Congruence Table on page #114
11 What’s the Difference between equality and congruence? A BAB represents the length AB, so you can think of AB as a variable representing a number.Helpful Hint
12 Geometric objects (figures / drawings) can be congruent to each other. CongruenceEqualityGeometric objects (figures / drawings) can be congruent to each other.Measurements (numbers)can be equal to each other.Statements use symbolStatements use = symbolNumbers are equal (=) andfigures are congruent ().Remember!
13 2.5 ApplicationWrite 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
14 The basic format of a two column proof: Page 115 Given - facts you are given to use.STARTING POINTProve – 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 givenThis is what you are asked to proveReasons
16 Application Statement Reason AB + BC = AC 2y + 3y – 9 = 21 5y – 9 = 21 PROVE: y = 6GIVEN:StatementReasonAB + BC = AC2y + 3y – 9 = 215y – 9 = 215y = 30y = 6Segment addition postulateSubstitutionCombine like termsAddition 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 = 139Prove : x = 43StatementReasonsm AOC = 139GivenM AOB + m BOC = m AOCAngle Addition Postulatex x = 139Substitution PropertySimplify or combine like terms3x + 10 = 1393x = 129Subtraction Property of Equalityx = 43Division Property of Equality
18 Using Properties to Justify Steps in Solving Equations Prove x = 20 and justify each step.Given: LM bisects KLNProve: x = 20StatementReasonsLM bisects KLNGivenMLN = KLM4x = 2x + 402x = 40x = 20Def of Angle BisectorSubstitution PropertySubtraction Property of EqualityDivision Property of Equality
19 Using Properties to Justify Steps in Solving Equations Now you trySolve for y and justify each stepGiven: AC = 21Prove : y = 6StatementReasonsAC = 21GivenAB + BC = ACSegment Addition Postulate2y + 3y - 9 = 21Substitution PropertySimplify5y – 9 = 215y = 30Addition Property of Equalityy = 6Division Property of EqualityFind AB and BC by substituting y = 6 into the expressions.
20 Using Properties of Equality and Congruence Name the property of congruence orequality the justifies each statement.a. K KReflective Property of Congruenceb. If 2x – 8 = 10, then 2x = 18Addition Property of Equalityc. If RS TW andTW PQ,then RS PQ.Transitive Property of Congruenced. If m A = m B, thenm B = m ASymmetric 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