How can you prove that the opposite angles in a cyclic quadrilateral are supplementary, and what implications does this property have when applied to problems involving incircles and excircles?

To prove that opposite angles in a cyclic quadrilateral are supplementary, consider a quadrilateral inscribed in a circle. By the Inscribed Angle Theorem, the measure of an angle is half the measure of the intercepted arc. Opposite angles intercept arcs that together sum to 360 degrees; thus, their measures sum to 180 degrees. This property implies that in problems involving incircles and excircles, the supplementary nature of opposite angles can help establish angle relationships and solve for unknowns.

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