The vertices of the triangle you lie on a circle with center o, angle aob is 80 degrees, arc AC:

The vertices of the triangle you lie on a circle with center o, angle aob is 80 degrees, arc AC: arc BC = 2: 3. Find the angles of the triangle ABC.

1) Since the angle <AOB is central, the arc ACB = 80 °, but the angle <ACB rests on the arc (360 ° – arc ACB) = 360 ° – 80 ° = 280 °.

2) Angle <ACB = 1/2 * 280 ° = 140 °, since this is an inscribed angle, and resting on this arc.

3) The remaining angles of the triangle are determined from the ratio: 2 * x + 3 * x + 140 ° = 180 ° (the sum of the angles in the triangle, and x is the unknown part in the proportion 2: 3).

4) Let’s solve the equation: 5 * x = 40 °; x = 40 ° / 5 = 8 °, whence <CAB = 2 * x = 2 * 8 ° = 16 °;

5) Angle <ABC = 3 * x = 3 * 8 ° = 24 °.

Answer: angles 140 °; 16 °; 24 °.