BIG IDEA: Reasoning and Proof ESSENTIAL UNDERSTANDINGS: Logical reasoning from one step to another is essential in building a proof. Logical reasoning.

Reasoning with Properties from

Algebra

BIG IDEA: Reasoning and ProofESSENTIAL UNDERSTANDINGS:

• Logical reasoning from one step to another is essential in building a proof.

• Reasons in a proof include given information, definitions, properties, postulates, and previously proven theorems.

MATHEMATICAL PRACTICE:Construct viable arguments and critique the reasoning of others

Follow the steps of the brainteaser using your

age. Then try it using a family member’s age. What do you notice? Explain how the brainteaser works.

Write down your age. Multiply it by 10. Add 8 to the product. Double that answer and then subtract 16. Finally, divide the result by 2.

GETTING READY

Let be real numbers. Addition Property: Subtraction Property: Multiplication Property: Division Property: Reflexive Property: Symmetric Property: Transitive Property: Substitution Property:

Distributive Property:

Algebraic Properties of Equality

, and a b c

If , then _____+_____=_____+_____a bIf , then _____ _____ _____ _____a b

If , then _____ _____=_____ _____a b If and 0, then _____ _____=_____ _____a b c

For any real number , _____=_____aIf , then _____=_____a b

If and _____=_____, then _____=_____a bIf , then can be ____________ for

in any equation or expression

a b a b

_____ _____a b c

a)

EX 1: What is the value of x? Justify each step.

b)

EX 1: What is the value of x? Justify each step.

4 2 3 5 11x x

The properties of equality reviewed in the

beginning of this lesson add to the students’ “toolbox” for writing proofs using deductive reasoning. It is important to make a connection between the algebraic proofs that you unknowingly completed throughout all of Algebra and the proofs that you will complete in Geometry. You should understand that the process of solving an equation is an algebraic proof, although the justifications for each step are not generally written down. Therefore, proofs are not unique to Geometry.

Math Background

Reflexive Property:

Symmetric Property:

Transitive Property:

Properties of equality are true for any numbers, while congruence properties are true for geometric figures

Properties of Congruence

AB AB

A A

If , then

If , then

AB CD CD AB

A B B A

If and , then

If and , then

If and , then

AB CD CD EF AB EF

A B B C A C

B A B C A C

a)

Example 2

Verify that AC BD AB CD

Statements Reasons

1. 1. Given

2. 2.

3. 3.

4. 4. Segment Addition Postulate

5. 5.

6. 6.

BC BC

AC AB BC

AB BC BC CD

AB CD

b) A baseball diamond is shown below. The

pitcher’s mound is at . Use the information to find the .

Example 2

34m

1 2 3 180

1 2 93

3 4 180

m m m

m m

m m

Proof: a convincing __________________ that uses

____________________ reasoning. A proof __________________ shows why a conjecture is __________

Two-column proof: lists each __________________ on the __________and the justification or ____________ for each statement is on the __________. Each statement must follow ______________________ from the __________ before it.

Proofs

Given: Prove:

Writing a Two-Column Proof

Statements Reasons

1. 1. Given

2. 2.

3. 3. Addition Property of Equality

4. 4.

5. 5.

2 2m m

1 3m m

m AEC m DEB

m AEC m DEB

1 2

3 2

m m m AEC

m m m DEB

Given: Prove:

Writing a Two-Column Proof

5 1 21x

4x

Statements Reasons

1. 1. Given

2. 2.

3. 3.

5 20x

2.4 p. 9910 – 14 all, 16 – 28 evens, 29 – 32 all, 36 – 45x3

20 questions