2.5 Reasoning in Algebra and

Geometry and 2.6 Proving Angles are

Congruent

Lesson Purpose

Objective

• To connect reasoning in algebra and geometry

Essential Question • How can you make a

conjecture and prove it is true?

ALGEBRAIC PROPERTIES OF EQUALITY

• Addition Property of Equality

• Subtraction Property of Equality

• Multiplication Property of Equality

• Division Property of Equality

• Substitution Property of Equality

• Transitive Property of Equality

• Reflexive Property of Equality

• Symmetric Property of Equality

• If a = b, then a + c = b + c.

• If a = b, then a – c = b – c.

• If a = b, then a c = b c.

• If a = b, then a/c = b/c.

• If a = b, then you may replace b with a in any expression.

• If a = b and b = c, then a = c.

• a = a

• If a = b, then b = a

Vocabulary Definitions

• Proof :

– Convincing argument that uses deductive reasoning

• Theorem:

– Is a conjecture or statement that you prove is true.

• Two Column Proof:

– Lists each statements on left and the proof on the right

Example: Justify Steps when Solving an Equation

Statements

• 206 = 11x + 96

• 206 - 96 = 11x + 96 - 96

• 110 = 11x + 0

• 110 = 11x

• 110/11=11x/11

• 10 = 1x

• 10 = x

• x = 10

Proof/Reasons

• Given

• Subtraction Property of Equality

• Subtraction

• Additive Identity Property

• Division Property of Equality

• Division

• Multiplicative Identity Property

• Symmetric Property

What is the value of x? Justify each step.

Given: AB bisects RAN

Statements:

• X= 2x-75

• x+75=2x

• 75=2x-x

• 75=x

Proof:

• Definition of an bisector

• Addition Property of Equality

• Subtraction Prop. Of equality

• Distribution Property

x (2x-75) R

A

N

B

Properties of Congruence

• Transitive Property of Congruence

• Reflexive Property of Congruence

• Symmetric Property of Congruence

• If a b and b c, then a c.

• If AB CD and CD EF, then AB EF.

• a a; AB BA

• If a b, then b a

• If AB CD, then CD AB.

Using Properties of Equality and Congruence

Statements

• A. ST ST

• B. If m R=m S and m S=m T, then m R=m T

• C. If AB=EF, then EF=AB

Reasons/Proof

• Reflexive Property of Congruence

• Transitive Property of Equality

• Symmetric of Equality

What is the name of the property that justifies the next step?

Statements

• A. AR TY

• TY AR

• B. 3(x+5)=9

• 3x+15=9

• C. ¼ x=7

• x=28

• D. m R =m R

Proof/Reasons

• Sym Property of

• Distributive Property

• Multiplication Property of Equality

• Reflective Property of Equality

Write a Two Column Proof

Given: AB CD

Prove AC BD

Statements:

• AB CD

• AB+BC=BC+CD

• AB+BC=AC or BC+CD=BD

• AC=BD

• AC BD

A B C D

• Proof:

• Given

• Addition Property of =

• Segment Addition Property

• Substitution Property of =

• Segments that = are

Proving Angles are Congruent

Vertical Angle Theorem

• Vertical Angles are congruent.

• JKL MKN

• JKM LKN

Theorem 2-4

• If 1 and 2 are right angles then 1 2.

1

2

Proving Angles are Congruent

Congruent Supplements theorem

• If 1 and 3 are supplements and 2 and

3 supplements, then 1 2.

Congruent Complements Theorem

• If 4 and 5 are complements and 5 and

6 complements then 4 6

1 2 3

4

5

6

Recap: Summary

• You use deductive reasoning and properties to solve equations and justify your reasoning.

• A proof is a convincing argument that uses deductive reasoning. A two column proof lists each statement on the left and the justification for each statement on the right.

Ticket Out

• What is the main difference between a property of equality and a property of congruence?