1. Warm Up: 1. Find the value of x. ANSWER 32
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.

2. 3.3 Proving Lines Parallel

3. Objectives
Prove that two lines are parallel based on given angle relationships

4. Corresponding Angles Converse Postulate
If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
If corr. s are , then lines are ║

5. Is there enough information in the diagram to conclude that m n
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, since corresponding angles are congruent then the lines are parallel.

6. Alternate Interior Angles Converse Theorem
If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel.
If alt. int. s are , then lines are ║

7. Two - Column Proof of the Converse of the Alternate Interior Angles Theorem3
Given: 1  2
Prove: m ║ n m 2 1 n
Statements:
1  2
2  3
1  3
m ║ n
Reasons:
Given
Vertical Angles are 
Transitive prop.
If corres. s are , then lines are ║

8. Consecutive Interior Angles Converse Theorem
If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel.
If cons. int. s are supp., then lines are ║

9. Proof of the Consecutive Interior Angles Converse Theorem
You are given that 4 and 5 are supplementary. By the LPP, 5 and 6 are also supplementary because they form a linear pair. If 2 s are supplementary to the same , then 4  6. Therefore, by the Converse of the Corresponding s Angles Postulate, g and h are parallel.
Given: 4 and 5 are supplementary Prove: g ║ h 6 g 5 4 h

10. Alternate Exterior Angles Converse Theorem
If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel.
If alt ext. s are , then lines are ║.

11. Same

12. Can you prove that lines a and b
are parallel? Explain why or why not.
Yes; Alternate Exterior Angles Converse.
ANSWER

13. Can you prove that lines a and b are parallel? Explain why or why not.
Yes; Corresponding Angles Converse.
ANSWER

14. m m 2 = 180°
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

15. Example 1: Determine which lines, if any, are parallel.
consecutive interior angles are supplementary. So,
consecutive interior angles are not supplementary. So, c is not parallel to a or b.
Answer:

16. Your Turn:
Determine which lines, if any, are parallel.
Answer:

17. Example 2: Alternate exterior angles Substitution
Find x and mZYN
so that
Solve
Alternate exterior angles
Substitution
Subtract 7x from each side.
Add 25 to each side.
Divide each side by 4.
Answer:

18. Your Turn:
Find x and mGBA
so that
Answer:

19. Example 3: ; 4.  7 + 6 = 180 4. Def. Suppl. s
Given:
Prove:
Reasons
Statements
1. Given 1. ;
2. Consecutive Interior Thm. 2.
3. Def. of congruent s 3.
4.  7 + 6 = Def. Suppl. s
5.  4 + 6 = Substitution
6.  4 and 6 are suppl 6. Def. Suppl. s
7. If cons. int. s are suppl., then lines are . 7.

20. Your Turn:
Given:
a || b
1  12
Prove: x || y

21. Daily Homework Quiz 1. 43
ANSWER 2.
Can you prove a b? If so,
what theorem would you use?
Yes; Alternate Interior Angle Converse
ANSWER

22. Daily Homework Quiz 3.
Which lines are parallel?
ANSWER
EF DG 4.
ANSWER
HJ MN by Transitive Property of Parallel Lines