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Warm Up

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Warm Up Answers

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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)

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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.

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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.

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**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.

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**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 = and = 3 are the same. Transitive Property of Equality If a = b and b = c, then a = c. EX: If = 7 and = 7, then = 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

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**2.5 Properties of Equality Table on page #113**

The Distributive Property states that a(b + c) = ab + ac. Remember!

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**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.

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**2.5 Properties of Congruence Table on page #114**

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**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

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**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!

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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

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**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

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**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

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**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)

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**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 x = 139 Substitution Property Simplify or combine like terms 3x + 10 = 139 3x = 129 Subtraction Property of Equality x = 43 Division Property of Equality

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**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

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**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.

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**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

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**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

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HOMEWORK COMPLETE 2-5 PACKET DUE THURSDAY NOV 1

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