1. Sec. 3-2 Proving Parallel Lines
Objective:
To use a Transversal in Proving Lines Parallel.
To relate Parallel & Perpendicular Lines.

2. Now we will work the proofs backwards.
In the last section we started with // lines and worked toward the angles.
In this section we will start with the angles and work towards the // lines.

3. Ways to Prove Two Lines Parallel
Show that corresponding angles are congruent.
Show that alternate interior angles are congruent.
Show that consecutive (same-side) interior angles are supplementary.
Show that consecutive (same-side) exterior angles are supplementary.
In a plane, show that the lines are perpendicular to the same line.
Lesson 2-5: Proving Lines Parallel

4. P(3-2) Converse of the Corresponding Angle Theorem
If two lines & a transversal intersect to form corresponding angles that are congruent then the two lines are //. 1 m
If 1  2, then m // n 2 n

5. Th(3-3) Converse of the Alternate Interior Angle Theorem
If two lines & a transversal intersect to form Alternate Interior that are congruent then the two lines are //.
3  6
4  5 3 4 5 6

6. Proof: Reasons Statements Given 3  6 3  1 2. Vertical s are 
Prove: n // m n 4 3 6 5 m 8 7
Reasons
Given
2. Vertical s are 
3. Subs.
4. If corresp s
are  then
lines are //.
Statements
3  6
3  1
3. 1  6
4. n // m

7. Th.(3-4) Converse of Same-Sided Interior Angle Theorem.
If two lines & a transversal intersect to form same - sided interior angles that are supplementary then the two lines are //. 3 4
m3 + m5 = 180 5 6
m4 + m6 = 180

8. Proof: Given: m3 + m5 = 180 Prove: n // m Statements7 m
Statements
1. m3 + m5 = 180
2. m5 + m7 = 180
3. m3 + m5 = m5 + m7
4. m3 = m7
5. 3  7
6. n // m
Reason
Given
 Add. Post.
Subs.
Subtr.
5. Def. of 
6. If corrsp. s are , then lines are //.

9. Th(3-5) If two lines are // to the same line, then they are // to each other.1 2 k 3 4 5 6 m 7 8 9 10 n 11 12

10. Th(3-5) If two lines are // to the same line, then they are // to each other.k 5 6 m 7 8 9 10 n 11 12

11. Th(3-5) In a plane, if 2 lines are perpendicular to the same line, then they are // to each other.

12. Corresponding Angles are  They are = 90 Alt. Int. s are 
Same-sided int. s are Supplementary
They are both = 90 t r s

13. Example 1: Solve for x and then solve for each angle such that n // m.
14 + 3(40) =
134 n
14 + 3x
5x - 66 m
14 + 3x = 5x -66
-3x x
14 = 2x – 66
80 = 2x 2
40 = x
5x – 66
5(40) – 66
134

14. Example 2: Find the m1 7x – 8 7(18) – 8 118 7x – 8 + 62 = 180

15. Ways to Prove Two Lines Parallel
Show that corresponding angles are congruent.
Show that alternate interior angles are congruent.
Show that consecutive (same-side) interior angles are supplementary.
Show that consecutive (same-side) exterior angles are supplementary.
In a plane, show that the lines are perpendicular to the same line.
Lesson 2-5: Proving Lines Parallel