Holt McDougal Geometry

2-5 Algebraic Proof

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

Objectives

Holt McDougal Geometry

2-5 Algebraic Proof

A proof uses logic, definitions, properties, and previously proven statements to “prove” that a conclusion is true.

An important part of writing a proof is giving justifications to show that every step is valid.

Possible ways to “prove” or justify each step:

1.A definition

2.A postulate

3.A property

4.Something given in the problem

Holt McDougal Geometry

2-5 Algebraic Proof

Holt McDougal Geometry

2-5 Algebraic Proof

The Distributive Property states that

a(b + c) = ab + ac.

Remember!

Holt McDougal Geometry

2-5 Algebraic Proof

Solve the equation 4m – 8 = –12. Write a justification for each step.

Example 1: Solving an Equation in Algebra

4m – 8 = –12

+8 +8

4m = –4

m = –1

Holt McDougal Geometry

2-5 Algebraic Proof

Check It Out! Example 1

t = –14

Solve the equation . Write a justification for each step.

Holt McDougal Geometry

2-5 Algebraic Proof

Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry.

A B

AB represents the length AB, so you can think of AB as a variable representing a number.

Helpful Hint

Holt McDougal Geometry

2-5 Algebraic Proof

Write a justification for each step.

Example 2: Solving an Equation in Geometry

NO = NM + MO

4x – 4 = 2x + (3x – 9)

4x – 4 = 5x – 9

–4 = x – 9

5 = x

Holt McDougal Geometry

2-5 Algebraic Proof

Check It Out! Example 2

Write a justification for each step.

x = 11

8x° = (3x + 5)° + (6x – 16)°

8x = 9x – 11

–x = –11

mABC = mABD + mDBC

Holt McDougal Geometry

2-5 Algebraic Proof

Check It Out! Example 2

Write a justification for each step.

x = 11

8x° = (3x + 5)° + (6x – 16)°

8x = 9x – 11

–x = –11

mABC = mABD + mDBC

Holt McDougal Geometry

2-5 Algebraic Proof

Classwork:

Exit Ticket with Partner

Homework:

•2.5 Practice B Homework Handout

• Must write out justification for each step to get credit!!

Holt McDougal Geometry

2-5 Algebraic Proof

Lesson Quiz: Part I

Solve each equation. Write a justification for each step.

1.

z – 5 = –12 Mult. Prop. of =

z = –7 Add. Prop. of =

Given

Holt McDougal Geometry

2-5 Algebraic Proof

Lesson Quiz: Part I

Solve each equation. Write a justification for each step.

1.

z – 5 = –12 Mult. Prop. of =

z = –7 Add. Prop. of =

Given

Holt McDougal Geometry

2-5 Algebraic Proof

Lesson Quiz: Part II

Solve each equation. Write a justification for each step.

2. 6r – 3 = –2(r + 1)

Given

6r – 3 = –2r – 2

8r – 3 = –2

Distrib. Prop.

Add. Prop. of =

6r – 3 = –2(r + 1)

8r = 1 Add. Prop. of =

Div. Prop. of =

Holt McDougal Geometry

2-5 Algebraic Proof

WARM UP

Give an example of the following properties:

1. Reflexive Property of Equality:

2. Symmetric Property of Equality:

3. Transitive Property of Equality:

Holt McDougal Geometry

2-5 Algebraic Proof

Objective:

Identify properties of equality and congruence.

Remember…..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.

Holt McDougal Geometry

2-5 Algebraic Proof

Holt McDougal Geometry

2-5 Algebraic Proof

Numbers are equal (=) and figures are congruent ().

Remember!

Holt McDougal Geometry

2-5 Algebraic Proof

Identify the property that justifies each statement.

A. QRS QRS

B. m1 = m2 so m2 = m1

C. AB CD and CD EF, so AB EF.

D. 32° = 32°

Example 1: Identifying Property of Equality and Congruence

Symm. Prop. of =

Trans. Prop of

Reflex. Prop. of =

Reflex. Prop. of .

Holt McDougal Geometry

2-5 Algebraic Proof

Check It Out! Example 2

Identify the property that justifies each statement.

A. DE = GH, so GH = DE.

B. 94° = 94°

C. 0 = a, and a = x. So 0 = x.

D. A Y, so Y A

Sym. Prop. of =

Reflex. Prop. of =

Trans. Prop. of =

Sym. Prop. of

Holt McDougal Geometry

2-5 Algebraic Proof

Lesson Review

Identify the property that justifies each statement.

1. x = y and y = z, so x = z.

2. DEF DEF

3. AB CD, so CD AB.

Trans. Prop. of =

Reflex. Prop. of

Sym. Prop. of

Holt McDougal Geometry

2-5 Algebraic Proof

Property Equality Example Congruence Example

Reflexive

Symmetric

Transitive

Classwork/Homework

come up with a unique example of each