A triangle is inscribed in a circle such that one side of the triangle is a diameter of the circle. What must be true about the angle…

This is Thales's theorem: an angle inscribed in a semicircle is a right angle. When one side of an inscribed triangle is a diameter, the opposite vertex lies on the circle, and the angle there intercepts a half circle of $180°$. An inscribed angle is half its intercepted arc, so the angle is $\frac{180°}{2} = 90°$. That is choice C. Choice A ($60°$) holds only for special triangles such as an equilateral one, which cannot have a diameter as a side. Choice B is impossible because the angle is exactly $90°$. Choice D is false because the angle never changes with the other side lengths.