Co-interior Angles
GeometryCo-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 are between the two parallel lines and on the SAME side of the transversal. Unlike alternate interior angles, they are NOT equal - they add up to $180^\circ$ instead. They form a "C" or "U" shape.
Example
If two parallel lines are cut by a transversal and one co-interior angle is $70^\circ$, the other co-interior angle on the same side is $110^\circ$. They are supplementary: $70 + 110 = 180$.
Key Insight
Co-interior angles are sometimes called "C angles" because the transversal and the parallel line segments form a C shape. The C shape is the visual key: same side, between the lines, adding to $180^\circ$.
Definition
Co-interior angles (same-side interior angles, or consecutive interior angles) lie between two lines on the same side of the transversal. Co-interior Angles Theorem: if two parallel lines are cut by a transversal, then co-interior angles are supplementary (sum to $180^\circ$). The converse is also true: if co-interior angles are supplementary, the lines are parallel.
Example
Parallel lines $l$ and $m$ cut by transversal $t$. On the right side of $t$, angle $4$ (below $l$) $= 65^\circ$ and angle $5$ (above $m$) $= 115^\circ$. Check: $65 + 115 = 180$ degrees (supplementary). If the lines were not parallel, this sum would not equal $180$.
Key Insight
While corresponding angles and alternate interior angles are equal for parallel lines, co-interior angles are supplementary (not equal). This is the one transversal relationship that involves addition rather than equality, making it the one students most often confuse.
Definition
Co-interior angles are supplementary when lines are parallel, which follows from the corresponding angles postulate: each co-interior angle and the corresponding angle of the other form a linear pair (supplementary), and the corresponding angles are equal, so the two co-interior angles sum to $180^\circ$. Formally, if $\alpha$ is a co-interior angle and $\beta$ is its alternate interior counterpart, then $\alpha + \beta = 180$ (since $\beta$ = its vertical angle, which is supplementary to $\alpha$).
Example
Proof: Let angle $4$ and angle $5$ be co-interior. Angle $4$ and angle $1$ are a linear pair (supplementary). Angle $1$ and angle $5$ are corresponding angles (equal, since lines parallel). Therefore angle $5$ = angle $1$ = $180$ - angle $4$, giving angle $4$ + angle $5$ = $180$.
Key Insight
The co-interior angle theorem is equivalent to the parallel postulate: Euclid's original fifth postulate can be stated as "if a transversal makes co-interior angles summing to less than $180^\circ$, the lines meet on that side." This direct connection to Euclid's fifth postulate makes co-interior angles central to the historical development of non-Euclidean geometry.