The points $M$ and $N$ are located respectively on the diagonal $(AC)$ and the side $(BC)$ of the square $ABCD$ such that $MN = MD$. Determine the measure of the angle $MDN$.
1. **Identify the given information and the geometric configuration:**
   - We have a square \(ABCD\) with points \(M\) and \(N\) such that \(M\) is on the diagonal \(AC\) and \(N\) is on the side \(BC\).
   - It is given that \(MN = MD\).

2. **Analyze the position of point \(M\):**
   - Since \(M\) is on the diagonal \(AC\) of the square, we can express \(M\) in terms of the coordinates of \(A\) and \(C\). Let the side length of the square be \(a\).
   - The coordinates of \(A\) are \((0, 0)\) and the coordinates of \(C\) are \((a, a)\).
   - Let \(M\) be at \((x, x)\) on the diagonal \(AC\).

3. **Analyze the position of point \(N\):**
   - Since \(N\) is on the side \(BC\), we can express \(N\) in terms of the coordinates of \(B\) and \(C\). The coordinates of \(B\) are \((a, 0)\) and the coordinates of \(C\) are \((a, a)\).
   - Let \(N\) be at \((a, y)\) on the side \(BC\).

4. **Use the given condition \(MN = MD\):**
   - Calculate the distance \(MN\):
     \[
     MN = \sqrt{(a - x)^2 + (y - x)^2}
     \]
   - Calculate the distance \(MD\):
     \[
     MD = \sqrt{(a - x)^2 + (a - x)^2} = \sqrt{2(a - x)^2} = \sqrt{2}(a - x)
     \]
   - Set the distances equal to each other:
     \[
     \sqrt{(a - x)^2 + (y - x)^2} = \sqrt{2}(a - x)
     \]

5. **Simplify the equation:**
   - Square both sides to eliminate the square roots:
     \[
     (a - x)^2 + (y - x)^2 = 2(a - x)^2
     \]
   - Simplify the equation:
     \[
     (a - x)^2 + (y - x)^2 = 2(a - x)^2
     \]
     \[
     (y - x)^2 = (a - x)^2
     \]
   - Take the square root of both sides:
     \[
     y - x = a - x \quad \text{or} \quad y - x = -(a - x)
     \]
   - Solve for \(y\):
     \[
     y = a \quad \text{or} \quad y = 2x - a
     \]

6. **Determine the valid solution:**
   - Since \(N\) is on the side \(BC\), \(y\) must be between 0 and \(a\). Therefore, \(y = a\) is the valid solution.
   - This means \(N\) is at \((a, a)\), which is point \(C\).

7. **Determine the measure of angle \(MDN\):**
   - Since \(N\) is at \(C\), the triangle \(MDN\) is actually triangle \(MDC\).
   - In a square, the diagonal \(AC\) divides the square into two 45-45-90 triangles.
   - Therefore, angle \(MDN\) is \(45^\circ\).

\(\blacksquare\)

The final answer is \( \boxed{ 45^\circ } \)