Co-Interior Angles
When a transversal crosses two
parallel
lines, the interior angles pair up in two ways. The
alternate interior
pairs sit on opposite sides and come out equal. The other pairing — the two
interior angles on the same side of the transversal — is what this page is about. They
are the co-interior angles (also called same-side interior,
allied, or "C" / "U" angles), and instead of being
equal they are supplementary: they add up to 180^\circ.
Trace the shape they make and you will see a C (or a U): the
two lines and the transversal enclose the pair on one side. Wherever you spot that "C", the two
angles inside it sum to a straight angle.
When a transversal cuts two parallel lines:
-
the two co-interior (same-side interior) angles add up to
180^\circ — they are supplementary;
-
they are the two angles inside the parallel lines and on the same side of
the transversal — the "C" shape;
-
so if one of them is known, the other is just
180^\circ minus it.
So, whenever a \parallel b, a co-interior pair satisfies:
\angle 4 + \angle 6 = 180^\circ \qquad\text{and}\qquad \angle 3 + \angle 5 = 180^\circ
Why it works
No measuring needed — two angle facts you already know do all the work. The goal is to show that
the co-interior pair \angle 4 and \angle 6
adds to 180^\circ. Step through the reason.
We borrowed the alternate-interior fact (\angle 4 = \angle 5) and the
straight-line fact (\angle 5 + \angle 6 = 180^\circ), and chained them:
\angle 4 + \angle 6 = \angle 5 + \angle 6 = 180^\circ.
The same argument on the left side gives \angle 3 + \angle 5 = 180^\circ.
So each "C" of co-interior angles is supplementary.
Practise: chase the angles
A fresh figure of two parallel lines cut by a transversal. Some angles are given; fill in every
other angle you can work out, ending with the highlighted one — using vertical angles, angles on
a line, and all the parallel-line angles (corresponding, alternate, and co-interior). Press
Refresh for a new one; Check explains each step.