Why is it that a quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary?

Answer 1

Because of one of the Circle Theorems that states that the angle subtended by any arc at the centre of the circle is half that at the circumference.

A rough version of the proof follows:

Suppose the quadrilateral ABCD is inscribed in a circle with centre O.

Join AO and CO. This partitions the circumference into two arcs - both AC but going around different sides of the centre.

One of the arcs AC, subtends angle B at the circumference and suppose the angle subtended by the same arc at O is X. Then 2B = X

The other arc AC subtends angle D at the circumference and suppose the angle subtended by the same arc at O is Y. Then 2D = Y

So 2B + 2D = X + Y

But X + Y = 360 degrees.

So B + D = 180 that is, B and D are supplementary.

And then, since A+B+C+D = 360, A + C = 180.

The converse can be proved similarly.