This is a proof that the sum of the measures of opposite angles in any simple cyclic quadrilateral is always 180°:
Let the polygon with vertices A,B,C, and D be a simple cyclic quadrilateral. Next, construct one of the quadrilateral's diagonals (for explanatory purposes, make it AC). Then, the sum of the arcs subtended by angles B and D will be (by definition) equal to the circumference of the circumscribed circle (360°). So the sum of angles B and D is 180°.
I was told that this proof isn't complete; that something is wrong with it. But I don't see what. What is wrong with this proof?