 # Chapter 2 Properties from Algebra

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Chapter 2 Properties from Algebra
Objective: To connect reasoning in Algebra & Geometry

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

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.

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)

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

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

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.

Example 1: Algebra Proof
3x = 15 x = 5 5 = x 1. Given Statement 2. Subtr. Prop 3. Division Prop 4. Symmetric Prop

Example 2 :  Addition Proof Given: mAOC = 139 Prove: x = 43
B x (2x + 10) C O Statements 1. mAOC = 139, mAOB = x, mBOC = 2x + 10 2. mAOC = mAOB + mBOC = 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.

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.

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.

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

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