1. Proving Lines Parallel
Chapter 3-5
Proving Lines Parallel

2. Lesson 3-5 Ideas/Vocabulary
Recognize angle conditions that occur with parallel lines.
Prove that two lines are parallel based on given angle relationships.
Lesson 3-5 Ideas/Vocabulary

3. Transitive property of Parallels
If two lines are parallel to the same line, then they are parallel to each other.
If p // q and q // r, then p // r. p q r

4. Reminders from Section 1
We will use these same theorems to prove the lines are parallel given certain angle information.

5. Corresponding Angle Theorem
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
// lines  corresponding s are 

6. Corresponding Angle Theorem

7. Alternate Interior Angle Theorem
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
// lines  Alt. Int. s are 

8. Alternate Interior Angle Theorem

9. Alternate Exterior Angle Theorem
If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.
// lines  Alt. Ext. s are 

10. Alternate Exterior Angle Theorem

11. Consecutive Interior Angle Theorem
If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.
// lines  Consec. Int. s are Supp.

12. Consecutive Interior Angle Theorem1 2
m1 + m2 = 180

13. Two  Theorem
If two lines are perpendicular to the same line, then they are parallel to each other.
If m  p and n  p, then m // n. p m n

14. Animation: Construct a Parallel Line Through a Point not on Line
Lesson 3-5 Postulates

15. Lesson 3-5 Theorems

16.  a//b  a is not // c  b is not // c 77o Identify Parallel Lines
Determine which lines, if any, are parallel.
Consec. Int. s are supp.
77o
 a//b
Alt. Int. s are not 
 a is not // c
Consec. Int. s are not supp.
 b is not // c
Lesson 3-5 Example 1

17. Determine which lines, if any are parallel. I. e || f II. e || g III
Determine which lines, if any are parallel. I. e || f II. e || g III. f || g A B C D
I only
II only
III only
I, II, and III
Lesson 3-5 CYP 1

18. Solve Problems with Parallel Lines
ALGEBRA Find x and m ZYN so that ||
Explore From the figure, you know that m WXP = 11x – 25 and m ZYN = 7x You also know that WXP and ZYN are alternate exterior angles.
Lesson 3-5 Example 2

19. ALGEBRA Find x and m ZYN so that || .
If Alt. Ext. angles are , then the lines will be //
m WXP = m ZYN Alternate exterior  thm.
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.
Lesson 3-5 Example 2

20. Solve Problems with Parallel Lines
Now use the value of x to find m ZYN.
m ZYN = 7x + 35 Original equation
= 7(15) + 35 x = 15
= 140 Simplify.
Answer: x = 15, m ZYN = 140
Lesson 3-5 Example 2

21. ALGEBRA Find x so that || .C D
x = 60
x = 9
x = 12
Lesson 3-5 CYP 2

22. Prove Lines Parallel
Prove: r || s
Given: ℓ || m
Lesson 3-5 Example 3

23. Prove Lines Parallel Proof: Statements Reasons 1. 1. Given
Consecutive Interior Angle Theorem
Definition of supplementary angles
Definition of congruent angles
Substitution
Definition of supplementary angles
If consecutive interior angles theorem
Lesson 3-5 Example 3

24. not enough information to determine
Given x || y and , can you use the Corresponding Angles Postulate to prove a || b? A B C
yes no
not enough information to determine
Lesson 3-5 CYP 3

25. Slope and Parallel Lines
Determine whether p || q.
slope of p:
slope of q:
Answer: Since the slopes are equal, p || q.
Lesson 3-5 Example 4

26. Determine whether r || s.A B C
Yes, r is parallel to s.
No, r is not parallel to s.
It cannot be determined.
Lesson 3-5 CYP 4