In the attached sketch,
for the cyclic quadrilateral abcd, Q+S=P+R=180 degrees.
If "d" is outside the circle on the line ad, angle S decreases, while Q remains the same. If "d" is inside the circle on the same line, S increases while Q remains the same.
In both these cases, Q+S is no longer 180 degrees.
Neither is P+R.
All angles sum to 360 degrees, but opposite angles will no longer sum to 180 degrees. Hence, if opposite angles sum to 180 degrees, the quadrilateral is cyclic, as a circle circumference will pass through all 4 vertices.