Equal corresponding angles mean parallel lines

The idea

If the corresponding angles formed by a transversal on a pair of lines are equal, the lines are parallel; if the lines are not parallel, the corresponding angles can never be equal.

Turn it around

Last time the lines were parallel, and the corresponding angles were equal. Now turn the question around.

A transversal crosses two lines, and the corresponding angles are equal. What do you expect about the two lines?

• They are parallel
• They meet somewhere
• We cannot tell

Turn line m

Line l stays fixed. Turn m and watch the four pairs of corresponding angles.

Only one place

Turning m changes the angles at the lower crossing, so the pairs match at one place only: when m runs the same way as l. There the lines are parallel and never meet. Anywhere else the angles differ, and the lines meet far away.

If the corresponding angles formed by a transversal on a pair of lines are equal, the lines are parallel. If the lines are not parallel, the corresponding angles can never be equal.

• Corresponding angles | The lines
• Equal | Parallel, they never meet
• Not equal | Not parallel, they meet

Notes

If the corresponding angles formed by a transversal on a pair of lines are equal, the lines are parallel. If the lines are not parallel, the corresponding angles can never be equal.

Check yourself

A transversal crosses two lines. ∠3 and ∠7 are corresponding angles and both are 65°. What can you say about the lines?

In the picture ∠1 = 118° and its corresponding angle ∠5 = 133°. What can you say about l and m?

The lines l and m are not parallel. Can a pair of corresponding angles be equal?

A transversal crosses two lines. ∠2 = ∠6 = 62°. Find ∠5.

Answer: 118 °

∠2 = ∠6, so the lines are parallel. ∠1 + ∠2 = 180°, so ∠1 = 118°. ∠5 corresponds to ∠1, so ∠5 = 118°.

• They meet on the right.. They would meet only if the corresponding angles were different.
• They are parallel. — correct. Yes! Equal corresponding angles mean the lines are parallel.
• They meet on the left.. They would meet only if the corresponding angles were different.
• They are parallel.. Parallel lines have equal corresponding angles. 118° and 133° are not equal.
• They are parallel, but not by much.. There is no "almost parallel". If the corresponding angles are not equal, the lines meet somewhere.
• They are not parallel. They meet on the right. — correct. The corresponding angles are not equal, so the lines are not parallel. Here m leans up to the right, so it meets l on the right.
• Yes, sometimes.. Equal corresponding angles only happen when the lines are parallel.
• No, never. — correct. Yes! If any pair of corresponding angles were equal, the lines would be parallel.
• Yes, if the transversal is long enough.. The length of the transversal does not matter.