From the image you uploaded, we are given a geometry problem in which MAN is a tangent to the circle at point A, and we are asked to find the angle ACB. Let me walk through the steps to solve this:

Step-by-Step Explanation:

1. Tangent-Radius Property: Since MAN is a tangent to the circle at A, and the line from the center of the circle to point A (which is the radius of the circle) is perpendicular to the tangent, we know that angle OAN is 90°.

2. Inscribed Angles and Chord: We are asked to find angle ACB, which is subtended by the chord AB. The inscribed angle theorem tells us that an angle subtended by a chord at the circumference is half of the angle subtended by the same chord at the center.

3. Given Angles: In the diagram, angle ADB is given as 80°, and angle BAD is given as 32°.

4. Key Geometry Rule: For any point on the circumference of a circle (in this case C), the angle subtended by the same arc (or chord) is half of the angle at the center. Therefore, angle ACB is half of angle AOB (which subtends the same chord).

Conclusion:

• Using the relationships between the angles, we can solve the problem and conclude that angle ACB is 40°.

Thus, the correct answer is:

B. 40°

Would you like more details on how to apply the inscribed angle theorem or have any other questions?

Here are 5 follow-up questions for deeper understanding:

1. How do you apply the inscribed angle theorem to other parts of the circle?
2. What is the relationship between the tangent and the radius of the circle at the point of contact?
3. How do angles subtended by the same chord relate to each other?
4. Can the angle in a semicircle ever be anything other than 90°?
5. How would this problem change if the circle had a different point of tangency?

Tip: Remember that angles in a triangle always add up to 180°, which is often helpful in geometry problems involving circles!