Cyclic Quadrilaterals

A cyclic quadrilateral is a four-sided shape whose four corners all sit on a single circle. That one condition forces a beautiful angle relationship: the two angles that face each other across the shape — its opposite angles — always add up to 180^\circ.

\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Angles that add to 180^\circ are called supplementary, so each pair of opposite angles in a cyclic quadrilateral is supplementary.

When all four vertices of a quadrilateral lie on a circle:

the quadrilateral is cyclic — every corner is on the circle;
its opposite angles add up to 180^\circ (they are supplementary);
the other pair of opposite angles adds to 180^\circ too;
an exterior angle of the quadrilateral equals the opposite interior angle.

Why it works

Four points dropped onto a circle and joined up always behave the same way. Step through it and watch the two opposite angles.

Whatever the angle at A turns out to be — call it x — the angle directly across at C is exactly 180^\circ - x, so together they make 180^\circ. The same is true of the other pair, B and D.