Co-interior Angles
Co-interior angles (also called same-side interior or consecutive interior angles) are between two lines on the same side of a transversal, summing to 180 degrees when the lines are parallel.
Formula
\text{angle } A + \text{angle } B = 180^\circ \text{ (when lines are parallel)}
Definition
Co-interior angles (also called same-side interior or consecutive interior angles) lie between two parallel lines on the SAME side of the transversal, forming a "C" or "U" shape; unlike alternate interior angles, they are NOT equal, they add up to $180^\circ$ instead (the Co-interior Angles Theorem, whose converse also holds). This follows from the corresponding angles postulate: each co-interior angle and the corresponding angle of the other form a linear pair, and since corresponding angles are equal, the two co-interior angles sum to $180^\circ$.
Example
If two parallel lines are cut by a transversal and one co-interior angle is $70^\circ$, the other on the same side is $110^\circ$ ($70+110=180$). For parallel lines $l$ and $m$ cut by transversal $t$: angle $4$ (below $l$) $=65^\circ$ and angle $5$ (above $m$) $=115^\circ$, and indeed $65+115=180$; if the lines were not parallel, this sum would not equal $180$.
Key Insight
Co-interior angles are sometimes called "C angles" because of the shape the transversal and parallel segments form: same side, between the lines, adding to $180^\circ$. While corresponding angles and alternate interior angles are equal for parallel lines, co-interior angles are the one relationship involving addition rather than equality, which is why students most often confuse it. Euclid's original fifth postulate can be stated in terms of co-interior angles, "if a transversal makes co-interior angles summing to less than $180^\circ$, the lines meet on that side," a direct historical link to the development of non-Euclidean geometry.