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The problem involves a right triangle \( BAC \), with angles labeled \( 1, 2, 3, 4, \) and \( 5 \) in the diagram. Let's analyze the situation step by step.

### Key Information:
1. \( \triangle BAC \) is a right triangle, with \( \angle B = 60^\circ \) (given) and \( \angle A = 90^\circ \) (since it's a right triangle).
2. \( \triangle AMC \) is isosceles, with \( \overline{AM} = \overline{MC} \).
3. \( AH \) is both the altitude and the median from vertex \( A \) to hypotenuse \( BC \).

### Step-by-step analysis:

1. **Angle 1 and Angle 2 (inside \( \triangle AHB \))**:
   - Since \( \angle AHB \) is part of \( \triangle BAC \), we know that \( \triangle AHB \) is also a right triangle.
   - \( \angle B = 60^\circ \), so the other non-right angle \( \angle 1 \) must satisfy \( 90^\circ - 60^\circ = 30^\circ \).
   - Since \( \triangle AHB \) is isosceles (because \( AH \) is both a median and an altitude), \( \angle 1 = \angle 2 = 30^\circ \).

2. **Angle 3 (the angle at vertex \( A \))**:
   - In \( \triangle AMC \), \( \overline{AM} = \overline{MC} \) (isosceles), and one of the base angles \( \angle AMC = 60^\circ \).
   - The vertex angle of the isosceles triangle, \( \angle 3 \), can be calculated as:
     \[
     \angle 3 = 180^\circ - 2 \times 60^\circ = 60^\circ
     \]

3. **Angle 4 (base angle of \( \triangle AMC \))**:
   - In \( \triangle AMC \), since it's isosceles, the base angles are equal, so \( \angle 4 = \angle 5 \).
   - From the fact that \( \angle AMC = 60^\circ \), we have:
     \[
     \angle 4 = \angle 5 = 60^\circ
     \]

Thus, the angles are:

- \( \angle 1 = 30^\circ \)
- \( \angle 2 = 30^\circ \)
- \( \angle 3 = 60^\circ \)
- \( \angle 4 = 60^\circ \)
- \( \angle 5 = 120^\circ \)

### Conclusion:
The correct option is **A**:
\[
\boxed{A}
\]

Let me know if you'd like further explanation or details!

### Related Questions:
1. What is the relationship between the angles in an isosceles triangle?
2. How do you calculate the angles of a triangle given two angles?
3. How does the altitude in a right triangle relate to the sides of the triangle?
4. What are the properties of medians in a triangle?
5. What is the significance of complementary angles in right triangles?

### Tip:
In any triangle, the sum of the interior angles is always \( 180^\circ \).

In the right triangle BAC, the altitude and the median are drawn from the right angle A. If triangle AMC is isosceles, what are the measures of angles 1, 2, 3, 4, and 5 marked in the figure?

Solve the problem of finding the measures of angles in a right triangle and an isosceles triangle. Understand the properties of angles, medians, and altitudes, along with detailed steps for solving this geometry problem. Suitable for students in Grades 9-10.

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