Opposite segments and cyclic quadrilaterals

A cyclic quadrilateral has all four vertices on one circle. Its opposite angles stand on the two complementary arcs between the same chord endpoints, so they total half a full turn: \(180^\circ\). This is the opposite-segments property.

Example

For cyclic quadrilateral \(PQRS\), if \(\angle PQR=100^\circ\), then \(\angle PSR=80^\circ\) (opposite angles of a cyclic quadrilateral). An exterior angle at \(S\), formed by extending one of its sides, is \(100^\circ\), equal to the opposite interior angle at \(Q\).

The exterior-angle result follows because both the exterior angle and the opposite interior angle are supplementary to the interior angle at \(S\). Do not assume any quadrilateral is cyclic: the question must give that all four vertices lie on one circle, or supply a justified route to that fact.

Choose a theorem from its conditions: centre and shared arc → double/half; diameter → right angle; same chord and same side → equal angles; opposite sides → supplementary; tangent at a radius endpoint → right angle. Include the matching geometrical reason in each deduction.