When a transversal crosses two parallel lines, it makes a four-way intersection at each crossing. Angles that sit in the same position at the two crossings — the same corner of each little intersection — are called corresponding angles. When the lines are parallel, each such pair is exactly equal.

They are the easiest pair to spot: slide one crossing along the transversal onto the other and the matching angles land on top of each other. Look for the letter F — the two parallel lines are its arms, the transversal its spine, and the corresponding angles tuck into the same corner of each arm.

Why it works: the sliding test

This is the starting fact of parallel-line geometry — the one we can't get from the earlier rules about a single crossing, because those say nothing about parallel lines. So instead of deriving it, we justify it by a movement you can picture. Step through it.

The slide carries the top crossing exactly onto the bottom one — only parallel lines let it land perfectly — so every corner keeps its angle. That is why all four corresponding pairs are equal, and it is the fact from which alternate and co-interior angles are then proved.

Practise: chase the angles

A fresh figure of two parallel lines cut by a transversal. Some angles are given; fill in every other angle you can work out, ending with the highlighted one — using corresponding angles, vertically opposite angles, and angles on a straight line. Press Refresh for a new one; Check explains each step.