In a circle, an inscribed angle that subtends a diameter is always a right angle (90°). This is due to the fact that the angle inscribed in a semicircle is always a right angle.

Given that $\overline{AB}$ is the diameter of circle $O$ and angle $C$ is inscribed, with point $C$ lying on the circle, the measure of angle $A$ (which is actually the measure of angle $\angle ACB$ because $\overline{AB}$ is the diameter) is:

$\angle A = 90^\circ$

This follows from the property of the circle that states an inscribed angle that subtends a semicircle is always a right angle.

Would you like further details or have any other questions?

Here are 8 relative questions to explore this topic further:

1. What is the definition of an inscribed angle in a circle?
2. Why does an inscribed angle subtending a semicircle always measure 90 degrees?
3. How can the inscribed angle theorem be used to find other angles in the circle?
4. What is the difference between a central angle and an inscribed angle in a circle?
5. How can you prove that an inscribed angle subtending a diameter is a right angle?
6. What are the properties of a diameter in relation to the circle's other chords and secants?
7. How do you calculate the arc length corresponding to a given central angle?
8. Can the property of the angle subtended by a diameter be generalized to other polygons inscribed in a circle?

Tip: Remember that in any circle, the angle subtended by a chord at the center is twice the angle subtended by the same chord at any point on the circle.