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The problem states that line segment \( AC \) is tangent to circle \( O \) at point \( A \), and the measure of arc \( \overarc{AB} = 59^\circ \). You are asked to find the measure of angle \( \angle ACB \).

### Solution:

1. **Tangent-Secant Angle Theorem:**
   The angle formed by a tangent and a chord (in this case, \( \angle ACB \)) is equal to half the measure of the intercepted arc.

   Here, the intercepted arc is \( \overarc{AB} \), which is given as \( 59^\circ \).

   According to the theorem:
   \[
   m\angle ACB = \frac{1}{2} \times m\overarc{AB}
   \]
   Substituting the given value:
   \[
   m\angle ACB = \frac{1}{2} \times 59^\circ = 29.5^\circ
   \]

2. **Rounding:**
   Since the available options are whole numbers, the closest option is \( 29^\circ \).

Thus, the measure of \( \angle ACB \) is approximately **29 degrees** (Option D).

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Would you like any further details or clarifications? Here are 5 related questions to expand your understanding:

1. How does the tangent-secant angle theorem apply in different configurations?
2. What is the relationship between central angles and their intercepted arcs?
3. Can you explain why the tangent line forms a 90° angle with the radius?
4. How would you calculate angles if the intercepted arc were larger?
5. How do inscribed angles relate to the arcs they intercept?

**Tip:** When working with tangent lines and circles, always remember that the radius to the point of tangency is perpendicular to the tangent.

AC is tangent to circle O at point A, and mAB = 59. What is m∠ACB?

Solve for angle ACB using the Tangent-Secant Angle Theorem with a given arc measure of 59°. Learn the relationship between tangents, arcs, and angles in this geometry problem suitable for Grades 9-12.

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