Question

Quadrilateral STWR is inscribed inside a circle as shown below. Write a proof showing that angles T and R are supplementary. Circle Q is shown with an inscribed quadrilateral labeled RSTW.

Answer 1

Consider the figure attached.

Let m(R)=α degrees and m(T)=β degrees.

1.
Angle R, is an inscribed angle, intercepting the arc WTS. 
This means that the measure of the arc WTS is double the measure of the angle R,

so m(arc WTS) = 2α degrees.

2.
Similarly, 

Angle T, is an inscribed angle, intercepting the arc WRS. So 

m(arc WRS) = 2β degrees.

3.
m(arc WTS)+m(arc WRS)=360° since these arcs cover the whole circle.

thus 

2α+2β=360°

divide by 2:

α+β=180°

this means T and R are supplementary angles.