Inscribed Quadrilateral

Prove that if a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Assume circle O has an inscribed quadrilateral, ABCD, where four inscribed angles are formed within circle O. In other words, chord AC creates two inscribed angles, ABC and CDB. Each angle equals half the measure of the arc intercepted by the angle (Theorem 6.9). The sum of the measure of the intercepted arcs of the opposite angles is the full circle or 360. That means the angles sum to half of this measure or 180. By Definition 4.14, this means that opposite angles are supplementary.