Question

Prove that two angles in the same segment of a circle are equal.

Answer 1

Let AB be the arc of the minor segment of a circle with center O.  Let  C and D be points on the major segment of the circle. 

Angles ∠ACB = 1/2 * ∠AOB  (angle made at the center by arc AB)
angle ∠ADB = 1/2 * ∠AOB.

Hence,  ∠ACB = ∠ADB