For the quadrilateral shown above, this conjecture says that:
Conjecture (Quadrilateral Sum ): Opposite angles in any quadrilateral inscribed in a circle are supplements of each other. (Their measures add up to 180 degrees.)
Arc(BCD)=2*Angle(A) and Arc(DAB)=2*Angle(C)
Arc(BCD)+Arc(DAB) = 360
2*Angle(A)+2*Angle(C) = 2*[Angle(A)+Angle(C)] = 360
Angle(A)+Angle(C) = 180
The proof is complete!