Reasoning with Properties of Algebra & Proving Statements About Segments CCSS: G-CO.12

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Essential Question(s) What algebra properties apply to angles and segments? How do we use properties of length and measure to justify segment and angle relationships? How do we justify statements about congruent segments?

Activator: Work with your partner. Make a list of Properties of Equality for Algebra. Give examples for each property. Solve writing down your reasoning for each step: 6x + 3 = 9(x -1).After you finish walk around to compare your results with the other groups.

Activator: Given: AB = BC Prove: AC = 2(BC) A B C

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

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 + 5 = x = 15 3 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 Statements 1. m  AOC = 139, m  AOB = x, m  BOC = 2x m  AOC = m  AOB + m  BOC = x + 2x = 3x = 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. x (2x + 10) A B O C

Example 3: Segment Addition Proof Given: AB = 4 + 2x BC = 15 – x AC = 21 Prove: x = 2 Statements 1.AB=4+2x, BC=15 – x, AC=21 2.AC = AB + BC 3.21 = 4 + 2x + 15 – x 4.21 = 19 + x 5. 2 = x 6.x = 2 Reasons 1.Given 2.Segment Add. Prop. 3.Subst. Prop. 4.Combined Like Term. 5.Subtr. Prop. 6.Symmetric Prop. A B C 15 – x 4 + 2x

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.

Theorem A true statement that follows as a result of other true statements.A true statement that follows as a result of other true statements. All theorems MUST be proved!All theorems MUST be proved!

2-Column Proof Numbered statements and corresponding reasons in a logical order organized into 2 columns.Numbered statements and corresponding reasons in a logical order organized into 2 columns. statementsreasons statementsreasons etc. etc.

Geometry Properties of Congruence 1.Reflexive Property: AB  AB  A   A 2.Symmetric Prop: If AB  CD, then CD  AB If  A   B, then  B   A 3.Transitive Prop: If AB  CD and CD  EF, then AB  EF IF  A   B and  B   C, then  A   C

Theorem 2.1- Properties of Segment Congruence Segment congruence is reflexive, symmetric, & transitive.

Proof of symmetric part of thm. 2.1 Statements 1. 2.AB = BC 3.BC = AB 4. Reasons 1.Given 2.Defn. of congruent segs. 3.Symmetric prop of = 4.Defn. of congruent segs.

Paragraph Proof Same argument as a 2-column proof, but each step is written as a sentence; therefore forming a paragraph. P X Y Q You are given that line segment PQ is congruent with line segment XY. By the definition of congruent segments, PQ=XY. By the symmetric property of equality XY = PQ. Therefore, by the definition of congruent segments, it follows that line segment XY congruent to line segment PQ.

Ex: Given: PQ=2x+5 QR=6x-15 PR=46 Prove: x=7 Statements 1.PQ=2x+5, QR=6x-15, PR= PQ+QR=PR 3. 2x+5+6x-15= x-10= x=56 6. x=7 Reasons 1.Given 2.Seg + Post. 3.Subst. prop of = 4.Simplify 5.+ prop of = 6.Division prop of = P QR

Ex: Given: Q is the midpoint of PR. Prove: PQ and QR = Statements 1.Q is midpt of PR 2.PQ=QR 3.PQ+QR=PR 4.QR+QR=PR 5.2QR=PR 6.QR= 7.PQ= Reasons 1.Given 2.Defn. of midpt 3.Seg + post 4.Subst. prop of = 5.Simplify 6.Division prop of = 7.Subst. prop

What did I learn Today? 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