Triangle ABC is inscribed in a circle with angle B = 82°, and C = 40°. Point M is the midpoint of arc AB. Determine the value of angle MAC.

Given triangle \( ABC \) inscribed in a circle, with \( \angle B = 82^\circ \) and \( \angle C = 40^\circ \), we need to find the value of \( \angle MAC \), where \( M \) is the midpoint of the arc \( AB \) (not containing \( C \)).

### Step 1: Determine \( \angle A \)
Since \( \triangle ABC \) is inscribed in a circle, the sum of the angles in the triangle is \( 180^\circ \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
\[
\angle A + 82^\circ + 40^\circ = 180^\circ
\]
\[
\angle A = 180^\circ - 122^\circ = 58^\circ
\]

### Step 2: Properties of the arc midpoint
Point \( M \) is the midpoint of arc \( AB \) that does not contain \( C \). A key property here is that the angle subtended by an arc at any point on the circle is half the angle subtended by the same arc at the center of the circle. Since \( M \) is the midpoint of the arc \( AB \), \( \angle AMB \) is a right angle \( 90^\circ \).

### Step 3: Calculate \( \angle MAC \)
To find \( \angle MAC \), we observe that \( M \), being the midpoint of the arc \( AB \), makes the angle \( \angle MAC \) half the measure of the arc \( AC \).

The arc \( AC \) subtends the angle \( \angle ABC \) at \( M \), and since \( M \) is the midpoint of the arc \( AB \), we subtract \( \angle C \) from \( 90^\circ \):

\[
\angle MAC = \angle MAB - \angle BAC
\]

Since \( M \) is the midpoint of the arc \( AB \), \( \angle MAB = \angle MAC = \frac{1}{2} (\angle ABC + \angle BCA) \).

\[
\angle MAC = \frac{1}{2} \times (90^\circ - 40^\circ) = 25^\circ
\]

Thus, the value of \( \angle MAC \) is \( 25^\circ \).

### Conclusion
The angle \( \angle MAC \) is \( \boxed{25^\circ} \).

Would you like further details or have any questions?

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Here are some related questions to consider:

1. How would the solution change if \( M \) were the midpoint of arc \( AC \)?
2. What is the relationship between the angles at the center and the circumference for any given arc?
3. Can you generalize the result for any triangle inscribed in a circle?
4. What would \( \angle MAC \) be if \( B \) were at the center of the circle?
5. How does the position of point \( M \) affect the angles in the triangle?
6. What other properties of circles and angles are used in problems like this?
7. If \( \angle B \) were \( 90^\circ \), what would \( \angle MAC \) be?
8. Can you use this approach to find any other angles in the triangle?

**Tip**: In circle geometry, always remember that angles subtended by the same arc are equal, and the angle subtended by a diameter is \( 90^\circ \).

Learn how to determine the angle MAC in triangle ABC inscribed in a circle with given angles B = 82° and C = 40°. Detailed step-by-step solution using circle geometry principles and angle properties. Suitable for high school students.

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