The AC side of triangle ABC passes through the center of the circle. Find ∠C if ∠A = 30 °.

Since AC passes through the center of the circle, this means the diameter.
The triangle ABC is inscribed in a circle and the center of the circle lies in the middle of AC, therefore the triangle ABC is rectangular with the hypotenuse AC (by the theorem on the circumscribed circle).
By the theorem on the sum of the angles of a triangle:
180 ° = ∠A + ∠B + ∠C
180 ° = 30 ° + 90 ° + ∠C
∠C = 60 °
Answer: 60