A triangle ABC is inscribed in a circle centered at point O. It is known that the angle AOC = 80 degrees

A triangle ABC is inscribed in a circle centered at point O. It is known that the angle AOC = 80 degrees, the angle AOB = 70 degrees. Find the degree measure of angle A.

By the condition of the problem, it is known that the triangle ABC is inscribed in a circle centered at the point O.

AOC angle is a central angle based on the AC arc.

Angle ABC is an inscribed angle based on the arc AC.

Therefore, ABC = 1/2 * AOC = 1/2 * 80 ° = 40 °.

Angle AOB is a central angle based on the arc AB.

Angle ACB is an inscribed angle resting on the arc AB.

Therefore, ACB = 1/2 * AOB = 1/2 * 70 ° = 35 °.

Since the sum of the angles of the triangle is 180 °, we have:

A = BAC = 180 ° – ABC – AСB = 180 ° – 40 ° – 35 ° = 105 °.