Points A, B, C lie on a circle centered at point O. The CAB angle is 60 °. The ratio of arcs AB

Points A, B, C lie on a circle centered at point O. The CAB angle is 60 °. The ratio of arcs AB and AC is 5: 3. Find the angles BOC and ABC.

The inscribed angle CAB = 60 and rests on the arc BC, then the degree measure of the arc BC = 2 * 60 = 120.

The sum of the degree measures of the arcs AC and AB is equal to: AC + AB = (360 – BC) = (360 – 120) = 240.

Let the degree measure of the arc AC = 3 * X0, then the arc AB = 5 * X0.

3 * X + 5 * X = 240.

8 * X = 240.

X = 240/8 = 30.

Then arc AC = 3 * 30 = 900, arc AB = 5 * 30 = 150.

The inscribed angle is equal to half of the degree measure of the arc on which it rests, then the angle ABC = 90/2 = 45, the central angle BOS is equal to the degree measure of the arc BC.

BOC angle = 2 * 60 = 120.

Answer: The BOC angle is 120, the ABC angle is 45.