Question

PROVE THAT THE MEASURE OF A CENTRAL ANGLE SUBTENDED BY AN ARC IS TWICE THE MEASURE OF AN INSCRIBED ANGLE IN THE CIRCLE SUBTENDED BY THE SAME ARC.

Answer 1

Answer: Angle AOC = 2 x Angle ABC

Explanation:

The diagram below is used to illustrate the proof

O is the center of the circle.

We want to prove that angle AOC = 2 x angle ABC

AO = BO = radius of circle

This means that triangle AOB is isosceles. In an isosceles triangle, the base angles are equal. Thus, angle OAB = angle OBA

Angle AOD is an exterior angle of triangle. According to the exterior angle theorem,

angle AOD = angle OAB + angle OBA = 2 x angle OBA

By applying the same theorem to triangle BOC,

angle COD = 2 x angle OBC

Thus,

angle AOD + angle COD = 2 x angle OBA + 2 x angle OBC

By factorizing the right side,

angle AOD + angle COD = 2(angle OBA + angle OBC)

Angle AOC = angle AOD + angle COD

Angle ABC = angle OBA + angle OBC

Thus, by substitution,

Angle AOC = 2 x Angle ABC