1. 3.33.3 Prove Lines are Parallel
Bell 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.

2. 3.3

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.

4. 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 75
o
also measures 75
o
since it
forms a linear pair with the 105o
angle. So,
corresponding angles are congruent and Postulate
16 says the lines are parallel.

5. 3.3

6. 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.

7. 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

8. 3.3Guided Practice
Can you prove that lines a and b are parallel?
Explain why or why not.
Yes; Corresponding Angles Converse
ANSWER
4.

9. 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°

10. 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

11. 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

12. 3.3

13. 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.

14. 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.

15. 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.

16. 3.3

17. 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.

18. 3.3
8. 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

19. 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 ConverseANSWER

20. 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

21. 3.3
Homework
Pg 169-173
#8, 14, 19, 29, 33