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**Chapter 2 Properties from Algebra**

Objective: To connect reasoning in Algebra & Geometry

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Objectives Review properties of equality and use them to write algebraic and geometric proofs. Identify properties of equality and congruence.

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**In Geometry you accept postulates & properties as true.**

You use Deductive Reasoning to prove other statements. In Algebra you accept the Properties of Equality as true also.

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**Algebra Properties of Equality**

Addition Property: If a = b, then a + c = b + c Subtraction Property: If a = b, then a – c = b – c Multiplication Property: If a = b, then a • c = b • c Division Property: If a = b, then a/c = b/c (c ≠ 0)

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**More Algebra Properties**

Reflexive Property: a = a (A number is equal to itself) Symmetric Property: If a = b, then b = a Transitive Property: If a = b & b = c, then a =c

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**2 more Algebra Properties**

Substitution Properties: (Subs.) If a = b, then “b” can replace “a” anywhere Distributive Properties: a(b +c) = ab + ac

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A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid.

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**Example 1: Algebra Proof**

3x = 15 x = 5 5 = x 1. Given Statement 2. Subtr. Prop 3. Division Prop 4. Symmetric Prop

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**Example 2 : Addition Proof Given: mAOC = 139 Prove: x = 43**

B x (2x + 10) C O Statements 1. mAOC = 139, mAOB = x, mBOC = 2x + 10 2. mAOC = mAOB + mBOC = x + 2x + 10 = 3x + 10 = 3x = x 7. x = 43 Reasons 1. Given 2. Addition Prop. 3. Subs. Prop. 4. Addition Prop 5. Subtr. Prop. 6. Division Prop. 7. Symmetric Prop.

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**Example 3: Segment Addition Proof Given: AB = 4 + 2x. BC = 15 – x**

Example 3: Segment Addition Proof Given: AB = 4 + 2x BC = 15 – x AC = 21 Prove: x = 2 A 15 – x C 4 + 2x B Statements AB=4+2x, BC=15 – x, AC=21 AC = AB + BC 21 = 4 + 2x + 15 – x 21 = 19 + x 2 = x x = 2 Reasons Given Segment Add. Prop. Subst. Prop. Combined Like Term. Subtr. Prop. Symmetric Prop.

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You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence.

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**Geometry Properties of Congruence**

Reflexive Property: AB AB A A Symmetric Prop: If AB CD, then CD AB If A B, then B A Transitive Prop: If AB CD and CD EF, then AB EF IF A B and B C, then A C

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**What did I learn Today? TU XY and XY AB, then TU AB Reflexive**

Name the property for each of the following steps. P Q, then Q P Symmetric Prop TU XY and XY AB, then TU AB Transitive Prop DF DF Reflexive

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