Angle Pairs and Parallel Lines

Where two lines cross, opposite angles are equal and adjacent ones add to 180°.

A transversal

A line crossing two parallel lines makes eight angles — but only two different sizes among them.

The names

Corresponding and alternate angles are equal. Co-interior angles add to 180°.

The shortcut worth having

Find one angle, and every other is either equal to it or its supplement.

Eight angles, two sizes

A transversal crossing two parallel lines makes eight angles, but only two distinct measures, and they add to 180°. Every angle is either equal to a given one or supplementary to it, which is why these problems collapse so quickly.

The names and what they claim

Corresponding angles are equal, alternate interior angles are equal, and co-interior angles are supplementary. Learning the names matters because a proof must cite the reason, not just assert the equality.

The converses prove parallelism

If corresponding angles are equal, the lines are parallel. This runs the other way and is how you prove lines parallel rather than assuming it. Confusing a statement with its converse is one of the standard errors in proof writing.

None of it holds without parallel lines

If the two lines are not parallel, all eight angles can differ. Every one of these relationships depends on the parallel condition, so a proof must establish parallelism before invoking them.