Question

Quadrilateral OPQR is inscribed inside a circle as shown below. Write a proof showing that angles O and Q are supplementary

Answer 1

As anoutline, angles O and Q are inscribed angles which are measured by one-half of its intercepted arc.
angle O = 1/2 arc RQP
angle Q = 1/2 arc ROP

so arcs RQP and ROP together make up the entire circle which has 360 degrees.

so angles O and Q is 1/2 (360) = 180 degrees (supplementary angles).