M
Math Coach Amy
Amy Ferguson Moncure

Geometry · Unit 3

Proving Lines Parallel

Use the converses of corresponding, alternate interior, alternate exterior, and consecutive interior angles.

Jump to section:

Key Ideas Easy Medium Hard

congruent corresponding / alt. interior / alt. exterior

supplementary consecutive interior

What to Remember

Converses

To prove two lines parallel, show one of these:

corresponding angles congruent, or

alternate interior angles congruent, or

alternate exterior angles congruent, or

consecutive interior angles supplementary.

Example: if corresponding angles are both 62°, the lines are parallel.

You may need a linear pair first

A given angle might not be the one you compare. Find its supplement or its vertical angle, then compare.

Example: a 130° angle has a 50° linear-pair partner. If the other line has a 50° corresponding angle, the lines are parallel.

Three lines

Check each pair. Maybe a ∥ b, or only a ∥ c, or none.

When they are not parallel

If corresponding angles are 70° and 75°, the converse does not apply. Do not claim the lines are parallel.

Level 1

Easy

Two lines, one clear pair.

Example
Lines r and s are cut by a transversal. A pair of corresponding angles measures 62° and 62°. Are r and s parallel? Why?

Corresponding angles are congruent.

Converse of corresponding angles: r ∥ s.

Answer: yes; corresponding angles converse

1
Lines r and s are cut by a transversal. Consecutive interior angles measure 115° and 65°. Are r and s parallel? Why?
Show Answer

115 + 65 = 180, so the consecutive interior angles are supplementary.

Converse of consecutive interior angles: r ∥ s.

Answer: yes; consecutive interior converse

2
Lines r and s are cut by a transversal. Alternate interior angles measure 48° and 48°. Are r and s parallel? Why?
Show Answer

Alternate interior angles are congruent.

r ∥ s by the alternate interior converse.

Answer: yes; alternate interior converse

3
Lines r and s are cut by a transversal. Corresponding angles measure 70° and 75°. Are r and s parallel?
Show Answer

70 ≠ 75, so corresponding angles are not congruent.

You cannot conclude that the lines are parallel.

Answer: no

4
Lines r and s are cut by a transversal. Alternate exterior angles measure 110° and 110°. Are r and s parallel? Why?
Show Answer

Alternate exterior angles are congruent.

r ∥ s.

Answer: yes; alternate exterior converse

5
Two lines are both perpendicular to the same third line. Must the two lines be parallel?
Show Answer

Yes. Each pair of corresponding angles is 90°.

Two lines perpendicular to the same line are parallel.

Answer: yes

Level 2

Medium

Find a hidden congruent or supplementary pair first.

Example
In the diagram, m∠2 = 125° and m∠5 = 55°.
Which lines, if any, are parallel? Name the converse you used.

∠2 and ∠4 are vertical, so m∠4 = 125°.

∠4 and ∠5 are consecutive interior.

125 + 55 = 180, so they are supplementary.

a ∥ b by the consecutive interior converse.

Answer: a ∥ b; consecutive interior converse

1
Three lines a, b, and c are cut by a transversal. The marked angles are 70° at a and 70° at b, in corresponding positions.
Which lines can you prove parallel?
Show Answer

Corresponding angles at a and b are congruent.

a ∥ b.

No information compares c, so you cannot prove a line through c is parallel.

Answer: a ∥ b only

2
Consecutive interior angles between r and s measure 88° and 92°. Are r and s parallel?
Show Answer

88 + 92 = 180, so they are supplementary.

r ∥ s.

Answer: yes

3
A transversal crosses lines p, q, and r. Alternate interior angles for p and q are 61° and 61°. Corresponding angles for q and r are 61° and 70°. Which lines are parallel?
Show Answer

p ∥ q by alternate interior converse.

q is not parallel to r: corresponding angles 61 and 70 are not congruent.

So only p ∥ q.

Answer: only p ∥ q

4
In the diagram, one marked angle is 118° and another, in the corresponding position on the next line, is 62°. Can you prove the lines parallel?
Show Answer

118 ≠ 62, so corresponding angles are not congruent.

118 + 62 = 180, but corresponding angles need to be congruent, not supplementary.

Do not conclude that the lines are parallel from this pair.

Answer: no

5
Lines a, b, and c are cut by a transversal. The marked angles are 55° at a and 125° at c, and those two angles are consecutive interior for a and c.
Which lines, if any, are parallel?
Show Answer

55 + 125 = 180, so consecutive interior angles for a and c are supplementary.

a ∥ c.

No marked pair involves b.

Answer: a ∥ c

Level 3

Hard

Three lines, mixed clues, or not enough information.

Example
Lines a, b, and c are cut by transversal t. The marked angles are 58° at b (upper-left) and 122° at c (upper-right).
Which lines, if any, can be proven parallel? State the converse.

Linear pair at c: 180 − 122 = 58°.

That 58° sits in the upper-left corner at c.

The marked angle at b is also 58° in the upper-left corner.

Corresponding angles are congruent.

b ∥ c by the corresponding angles converse.

Answer: b ∥ c; corresponding angles converse

1
A student marks 80° and 80° that are vertical, not corresponding, on two lines cut by a transversal. Can the student prove the lines parallel from that pair alone?
Show Answer

Vertical angles are always congruent, even when the lines are not parallel.

That pair gives no information about whether the two lines are parallel.

No.

Answer: no

2
Lines a, b, and c are cut by a transversal. Corresponding angles for a and b are 64° and 64°. Consecutive interior angles for b and c are 64° and 116°. Which lines are parallel?
Show Answer

a ∥ b (corresponding converse).

64 + 116 = 180, so b ∥ c (consecutive interior converse).

Parallel is transitive: a ∥ c too.

All three are parallel.

Answer: a ∥ b ∥ c

3
In a diagram with three lines and one transversal, two marked angles are 50° and 140°. The 50° is corresponding to an unmarked angle that is a linear pair with the 140°. Which lines are parallel?
Show Answer

Linear pair with 140° is 40°.

Corresponding would need 50 = 40.

50 ≠ 40, so corresponding angles are not congruent.

You cannot prove the lines parallel.

Answer: none

4
Lines p and q are cut by two different transversals. One transversal makes corresponding angles 72° and 72°. The other transversal makes corresponding angles 81° and 81°. What can you conclude?
Show Answer

Either transversal’s corresponding pair is enough.

p ∥ q.

The second pair matches because the lines are already parallel; it is consistent, not a second conclusion.

Answer: p ∥ q

5
To prove a ∥ b, you know m∠4 = 3x + 10 and m∠6 = 5x − 6, and 4 and 6 are alternate interior. Find x, then say whether that value is enough to prove the lines parallel.
Show Answer

Set them equal to use the converse: 3x + 10 = 5x − 6.

10 + 6 = 5x − 3x

16 = 2x

x = 8

Both angles equal 3(8) + 10 = 34°.

Alternate interior angles are congruent, so a ∥ b.

Answer: x = 8; a ∥ b

Need help proving two lines are parallel?

Book a Session with Amy