Question

Quadrilateral EFGH is inscribed inside a circle as shown below. Write a proof showing that angles H and F are supplementary.

Circle J is shown with an inscribed quadrilateral labeled EFGH.

Answer 1

That is called a cyclic quadrilateral. Angles E and G add up to 180 degrees and angles H and F add up to 180 degrees.

Answer 2

Explanation:

Given is a quadrilateral EFGH inscribed in a circle with centre J.

Consider angle E, which is subtended by minor arc FGH.

Hence angle E = half of angle FGH (by subtended angles theorems for circles)

Similarly G is the angle subtended by major arc FGH

angle G = half of angle subtended by major arc FEH

Since total angle around J is 360 degrees we have

angle subended by minor and major arc add to 360

Consequently, angle E + angle G = 1/2 (360) = 180

Hence E and G are supplementary