Reasoning with Properties Reasoning with Properties of Algebra &of Algebra &

Proving Statements About Proving Statements About SegmentsSegments

CCSS: G-CO.12CCSS: G-CO.12

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

Identify properties of equality and Identify properties of equality and congruence.congruence.

ObjectivesObjectives

• 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 = 20

-5 -5

3x = 15

3 3

x = 5

5 = x

1. Given Statement

2. Subtr. Prop

3. Division Prop

4. Symmetric Prop

Example 2 : Addition Proof

Given: mAOC = 139Prove: x = 43

Statements1. mAOC = 139, mAOB = x,

mBOC = 2x + 10

2. mAOC = mAOB + mBOC 3. 139 = x + 2x + 104. 139 = 3x + 105. 129 = 3x6. 43 = x7. x = 43

Reasons1. Given2. Addition Prop.3. Subs. Prop.4. Addition Prop5. Subtr. Prop.6. Division Prop.7. Symmetric Prop.

x(2x + 10)

AB

O

C

Example 3: Segment Addition Proof

Given: AB = 4 + 2x BC = 15 – x

AC = 21Prove: 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 C15 – 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.

TheoremTheorem

• A true statement that follows as a result of A true statement that follows as a result of other true statements. other true statements.

• All theorems MUST be proved!All theorems MUST be proved!

2-Column Proof2-Column Proof

• Numbered statements and corresponding Numbered statements and corresponding reasons in a logical order organized into 2 reasons in a logical order organized into 2 columns.columns.

statementsstatements reasonsreasons

1.1. 1.1.

2.2. 2.2.

3.3. 3.3.

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.

AB.AB AB,any For

AB.BC then BC,AB If

CD.AB then CD,BC and BCAB If

Proof of symmetric part of thm. Proof of symmetric part of thm. 2.12.1

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.

BCAB

ABBC

Paragraph ProofParagraph 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=7Statements

1. PQ=2x+5, QR=6x-15,

PR=46.

2. PQ+QR=PR

3. 2x+5+6x-15=46

4. 8x-10=46

5. 8x=56

6. x=7

Reasons

1. Given

2. Seg + Post.

3. Subst. prop of =

4. Simplify

5. + prop of =

6. Division prop of =

P

Q

R

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

2

PR

2

PR

2

PR

What did I learn Today?

• Name the property for each of the following steps.

P Q, then Q PSymmetric Prop

• TU XY and XY AB, then TU ABTransitive Prop

• DF DFReflexive