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Proving Lines Parallel 3-4C

- p. 207

You found slopes of lines and used them to

identify parallel and perpendicular lines.

- Recognize angle pairs that occur with parallel

lines.

- Prove that two lines are parallel.

Are these lines parallel?

- Is there information in this figure to assume

that the lines are parallel?

What information from the last section can we

use to show that these lines are parallel. What

do we need to add first?

Are these lines parallel?

t

a

b

If a transversal is added to the picture, what

information would tell you these lines are

parallel. (There are four different things you

can pick.)

Angles that can be used to show parallel lines

- Alternate exterior angles
- Alternate interior angles
- Corresponding angles
- Same side interior angles that add to 180

List the pair of parallel lines. How do you know

the lines are parallel?

a ll b by alt. int. angles are congruent

b ll c by corresponding angles are congruent

a ll c by alt. int. angles are congruent

Converse of Corresponding Angles Postulate

If two lines are cut by a transversal so that a

pair of corresponding angles are congruent, then

the lines are parallel.

Page 207

Euclids Postulate

- This is one of Euclids five original postulates

from 300 BC. Postulate 2.1 and Theorem 2.10 are

similar to two of Euclids other postulates.

Page 208

Page 208

A contractor wants to guarantee that the new

street she is marking of is parallel to Douglass

Street. She finds that the measure of angle 1 is

45. Give three different ways that she can use

angles to be sure that the new street is parallel

to Douglass Street.

Main St.

Douglass St.

1

2

3

4

New St.

5

- A. Given ?1 ? ?3, is it possible to prove that

any of the lines shown are parallel? If so, state

the postulate or theorem that justifies your

answer.

?1 and ?3 are corresponding angles of lines a and

b.

Answer Since ?1 ? ?3, ab by the Converse of

the Corresponding Angles Postulate.

- B. Given m?1 103 and m?4 100, is it

possible to prove that any of the lines shown are

parallel? If so, state the postulate or theorem

that justifies your answer.

?1 and ?4 are alternate interior angles of lines

a and c.

Answer Since ?1 is not congruent to ?4, line a

is not parallel to line c by the Converse of the

Alternate Interior Angles Theorem.

A. Given ?1 ? ?5, is it possible to prove that

any of the lines shown are parallel?

A. Yes l n B. Yes m n C. Yes l m D. It

is not possible to prove any of the lines

parallel.

Read the Test Item From the figure, you know

that m?WXP 11x 25 and m?ZYN 7x 35. You

are asked to find m?ZYN.

Solve the Test Item ?WXP and ?ZYN are alternate

exterior angles. For line PQ to be parallel to

line MN, the alternate exterior angles must be

congruent. So m?WXP m?ZYN. Substitute the given

angle measures into this equation and solve for

x. Once you know the value of x, use substitution

to find m?ZYN.

11x 25 7x 35 Substitution 4x 25

35 Subtract 7x from each side. 4x 60 Add 25 to

each side. x 15 Divide each side by 4.

Now use the value of x to find m?ZYN.

m?ZYN 7x 35 Original equation

7(15) 35 x 15 140 Simplify.

Answer m?ZYN 140

How can we show (prove) that lines are parallel?

- By showing that alternate exterior angles,

alternate interior, corresponding angles or

same-side interior angles are congruent.

3-5 Assignment

- Page 211, 6, 8-20 even