Recall that an inscribed (or 'cyclic') quadrilateral is one where the four vertices all lie on a circle. It turns out that the interior angles of such a figure have a special relationship. Each pair of opposite interior angles are supplementary - that is, they always add up to 180°.
In the figure above, drag any vertex around the circle. Note how the blue and red pairs of angles always add to 180°.
In the figure above, if you drag a point past its neighbor the quadrilateral will become 'crossed' where one side crossed over another. In such 'crossed' quadrilaterals the interior angle property no longer holds. (Most properties of polygons are invalid when the polygon is crossed). The angles instead become congruent (equal in measure).
In the figure above