the parallel-lines criterion
How can you be sure two lines are truly parallel — never meeting, no matter how far you extend them — without travelling to infinity to check? The answer is to read the angles a transversal makes with them. The parallel-lines criterion is the package of theorems that turns an angle equality at a single crossing into a guarantee of parallelism, and back again.
Precisely, for two lines cut by a transversal, the following four statements are all equivalent, and any one of them holds if and only if the lines are parallel: corresponding angles are equal; alternate interior angles are equal; alternate exterior angles are equal; co-interior (same-side interior) angles are supplementary (add to 180). To prove two given lines parallel, you exhibit just one of these — for example, show one pair of alternate interior angles is equal — and parallelism follows. To use parallelism, you run the implication the other way and read off the angles. The whole web is held together by linear pairs and vertical angles, so any one relation forces the rest.
An honest caveat: this criterion belongs to Euclidean geometry, where it rests on the parallel postulate. In non-Euclidean geometries the postulate is dropped, and lines can fail to meet without the angle relations holding in this neat way. So 'equal corresponding angles means parallel' is a theorem of the Euclidean plane, not a universal truth of all geometry — a hint of the deep choice hiding behind the fifth postulate.
You measure a pair of alternate interior angles where a transversal crosses two lines and both read 58 degrees. By the criterion (equal alternate interior angles), the two lines are parallel — you never had to extend them to check.
One equal angle pair at a crossing certifies parallelism.
Remember co-interior angles are the exception: for them parallelism means supplementary (sum 180), not equal. And the whole criterion is a Euclidean theorem resting on the parallel postulate.