An isosceles triangle ABC with base AC is inscribed in a circle with center O. Angle AOC = 112.

An isosceles triangle ABC with base AC is inscribed in a circle with center O. Angle AOC = 112. Find the angles of triangle ABC.

1. The inscribed angle ABC is equal to half of the corresponding central angle AOC resting on the same arc

∠B = ∠ABC = 1/2 * ∠AOC = 1/2 * 112 ° = 56 °.

2. In an isosceles triangle ABC, the angles at the base AC are:

∠A = ∠C.

3. The sum of the angles of the triangle is 180 °:

∠A + ∠B + ∠C = 180 °;
∠A + 56 ° + ∠C = 180 °;
∠A + ∠C = 180 ° – 56 °;
∠A + ∠C = 124 °;
∠A = ∠C = 1/2 * 124 ° = 62 °.
4. The angles of triangle ABC are equal:

∠A = 62 °;
∠B = 56 °;
∠C = 62 °.
Answer: 62 °; 56 °; 62 °.