co-interior angles

Among the angles a transversal makes with two lines, two of the interior angles sit on the same side of the transversal — one at the upper crossing and one at the lower, both leaning toward the same edge. These are the co-interior angles. They are sometimes called the C-angles because the two angles open toward each other like the inside of a letter C, or same-side interior angles, which says exactly what they are.

Precisely: pick the two interior angles that lie on the same side of the transversal (say both on its left). The co-interior-angles theorem says that if the two crossed lines are parallel, then a pair of co-interior angles is supplementary — they add to 180, not 180-each but together. The reason: one of them equals the alternate interior angle of the other (which is equal under parallelism), and that alternate angle forms a linear pair with the remaining co-interior angle, giving a sum of 180. The converse holds too: if co-interior angles add to 180, the lines are parallel.

The trap here is the most common in the whole transversal picture: co-interior angles are supplementary, not equal. Corresponding and alternate angles are equal under parallelism; co-interior angles are the odd family out, summing to 180. Mixing these up turns many a parallel-line proof inside out, so always ask first which family you are looking at.