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

Explanation:

Arc STR measures twice the measure of angle R, and arc WRS measures twice the measure of angle T.

STR = 2 x ∠R and WRS = 2 x ∠T --- This is because of the Inscribed Angle Theorem

If the measure of arcs STR and WRS are added together, the total would be 360° since a full circle is made up of 360°.

So mSTR + mWRS = 360°

Substituting the angle measures of R and T in for the arcs, the equation becomes:

2 • ∠R + 2 • ∠T = 360°

Simplifying further:

2 • [∠R + ∠T] = 360°

∠R + ∠T = 360° / 2

∠R + ∠T = 180° - This proves that opposite angles of a quadrilateral inscribed in a circle are supplementary.

Answer 2

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.