Inscribed Angles

An inscribed angle is an angle with its vertex on the circle and whose sides are chords of the circle:

When a side of an inscribed angle contains a diameter of the circle, it is easy to see that the measure of the inscribed angle is half the measure of its intercepted arc:

 because triangle AOB is isosceles (BO and AO are radii).  But

 because an external angle of a triangle is the sum to the two remote interior angles.  Therefore .  But the measure of arc AC is the same as the measure of the central angle, AOC.

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