Cho Hình 15. Tính \( \hat{A_3}, \hat{A_4}? \) Tính \( \overline{MN}. \)

To solve the problem related to Figure 15, we'll need to find \(\hat{A_3}\) and \(\hat{A_4}\), and then compute the length of \(\overline{MN}\).

### Given:
- \(\hat{MAY} = 49^\circ\)
- \(\hat{SAY} = 55^\circ\)
- \(\hat{MAZ} = \hat{AZN}\)

### Step 1: Calculate \(\hat{A_3}\) and \(\hat{A_4}\)

We know that the sum of angles around point A is \(180^\circ\).

1. Find \(\hat{A_3}\):
\[
\hat{A_3} = 180^\circ - (\hat{MAY} + \hat{SAY}) = 180^\circ - (49^\circ + 55^\circ) = 180^\circ - 104^\circ = 76^\circ
\]

2. Find \(\hat{A_4}\):
Since \(\hat{MAZ} = \hat{AZN}\) and they form a linear pair with \(\hat{A_3}\),
\[
\hat{A_4} = 180^\circ - \hat{A_3} = 180^\circ - 76^\circ = 104^\circ
\]

### Step 2: Calculate \(\overline{MN}\)

Assuming \(\overline{MN}\) is a segment of a triangle rest on angles we calculated, if we require lengths, we need more information such as either side lengths or additional angles from the triangle.

If you have the lengths of the sides or specific ratios or coordinates, please provide so we can calculate the exact length of \(\overline{MN}\).

### Summary:
- \(\hat{A_3} = 76^\circ\)
- \(\hat{A_4} = 104^\circ\)
- For \(\overline{MN}\), further details are needed.