3.33.3 Prove Lines are ParallelBell Thinger

2. Write the converse of the following statement. If it is raining, then Josh needs an umbrella.

ANSWER If Josh needs an umbrella, then it is raining.

ANSWER 32

1. Find the value of x.

3.3

3.3Example 1

ALGEBRA

Find the value of x that makes m n.

SOLUTION

Lines m and n are parallel if the marked corresponding angles are congruent.

3x = 60

x = 20

The lines m and n are parallel when x = 20.

(3x + 5)o = 65o Use Postulate 16 to write an equation

Subtract 5 from each side.

Divide each side by 3.

3.3Guided Practice

1. Is there enough information in the diagram to conclude that m n? Explain.

ANSWER

Yes. m n because the angle corresponding to the angle measuring 75o also measures 75o since it forms a linear pair with the 105o angle. So, corresponding angles are congruent and Postulate 16 says the lines are parallel.

3.3

3.3Example 2 How can you tell whether the sides of the pattern are parallel in the photo of a diamond-back snake?

Snake Patterns

SOLUTION

Because the alternate interior angles are congruent, you know that the sides of the pattern are parallel.

3.3Guided Practice

Can you prove that lines a and b are parallel? Explain why or why not.

3.

Yes; Alternate Exterior Angles Converse

ANSWER

3.3Guided Practice

Can you prove that lines a and b are parallel? Explain why or why not.

Yes; Corresponding Angles Converse

ANSWER

4.

3.3Guided Practice

Can you prove that lines a and b are parallel? Explain why or why not.

No; Supplementary angles do not have to be congruent.

ANSWER

5. m 1 + m 2 = 180°

3.3Example 3

Prove that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.

SOLUTION

GIVEN : ∠ 4 ∠ 5

PROVE : g h

3.3Example 3

g h4. 4. Corresponding Angles Converse

STATEMENTS REASONS

1. Given1. 4 5

2. 1 4 2. Vertical Angles Congruence Theorem

3. 1 5 3. Transitive Property of Congruence

3.3

3.3Example 4

SOLUTION

Look at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time.

In the figure, r s and 1 is congruent to 3. Prove p q.

3.3Example 4

Plan for Proof

a. Look at 1 and 2. Look at 2 and 3.b.

1 2 because r s. If 2 3 then p q.

3.3Example 4

Plan in Action

a. It is given that r s, so by the Corresponding Angles Postulate, 1 2.

b. It is also given that 1 3. Then 2 3 by the Transitive Property of Congruence for angles. Therefore, by the Alternate Interior Angles Converse, p q.

3.3

3.3Example 5

The flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe.

U.S. Flag

SOLUTION

The stripes from top to bottom can be named s1, s2, s3, . . . , s13. Each stripe is parallel to the one below it, so s1 s2, s2 s3, and so on. Then s1 s3 by the Transitive Property of Parallel Lines. Similarly, because s3 s4, it follows that s1 s4. By continuing this reasoning, s1 s13. So, the top stripe is parallel to the bottom stripe.

3.38. Each step is parallel to the step

immediately above it. The bottom step is parallel to the ground. Explain why the top step is parallel to the ground.

ANSWER

All of the steps are parallel. Since the bottom step is parallel to the ground, the Transitive Property of Parallel Lines applies, and the top step is parallel to the ground.

Guided Practice

3.3Exit Slip

1. Find the value of x that makes p q.

43ANSWER

Can you prove a b? If so,what theorem would you use?

2.

Yes; Alternate Interior Angle Converse ANSWER

3.3Exit Slip

3. Which lines are parallel?

EF DGANSWER

4. In the figure, if HJ KL and KL MN,what can you conclude?What theorem justifies your conclusion?

HJ MN by Transitive Property of Parallel Lines

ANSWER

3.3

Homework

Pg 169-173 #8, 14, 19, 29, 33