In a circle with center O, three radii OB, OС, OA are drawn, angle AOB = angle BC. prove that the angle OАВ = angle OСВ.

Given:

Circle (O, r);

r = AO = OB = OC;

AOB = BOC;

Prove: OAB = OCB;

Proof:

1.Consider triangles COB and BOA:

CO = OA; (by condition CO and OA are equal to the radius of the circle)

OB = OB (common side)

angle AOB = angle BOC (as required)

So COB = BOA (on both sides and the angle between them)

2. The angles OAB and OCB are corresponding (in equal triangles), which means they are equal.