Find the value of the inscribed angle α resting on the chord AB, equal to the radius of the circle.

Draw the radii ОА and ОВ to the ends of the chord AB.

Then in triangle OAB the side OA = OB = AB = R, and therefore the triangle OAB is equilateral, and then all its internal angles are 60. The central angle AOB = 60, then the degree measure of the arc AB is the same 60.

The inscribed angle ACB rests on the arc AB = 60, then the value of the inscribed angle is equal to half of the degree measure of the arc on which it rests. Angle ACB = 60/2 = 30.

Answer: The angle α is equal to 30.