To solve this problem, let's first understand the diagram and question.

The problem states that $AC$ is the diameter of circle $D$, and asks for the sum of $m \angle A$ and $m \angle C$. We can apply properties of circles and inscribed angles.

Key Points:

• $AC$ is the diameter of the circle.
• In any circle, an angle inscribed in a semicircle (formed by a diameter) is a right angle (90°). This is a result of the inscribed angle theorem.

Applying the Theorem:

• Since $\overline{AC}$ is the diameter, $\angle ABC$ (an inscribed angle formed with the diameter as one side) must be 90°.
• Therefore, $\angle A$ and $\angle C$ are angles in a right triangle where $\overline{AB}$ and $\overline{BC}$ are the other two sides.

Conclusion:

In any right triangle, the two non-right angles must sum to 90°. Hence, $m\angle A + m\angle C = 90^\circ.$

Would you like further clarification or more details?

Related Questions:

1. What is the inscribed angle theorem?
2. Can the sum of angles in a triangle exceed 180°?
3. How do you prove that an angle inscribed in a semicircle is a right angle?
4. What is the relationship between the central and inscribed angles subtended by the same arc?
5. What happens if $\overline{AC}$ is not the diameter?

Tip:

In any triangle, the sum of the three interior angles always equals 180°.