We are given a square $ABCD$ with points $M$ and $N$ lying on segments $AD$ and $DC$, respectively. The angles $B\hat{M}A = N\hat{M}D = 60^\circ$, and we are tasked with finding $M\hat{B}N$.

Step 1: Analyze the Geometry

Let's start by considering the geometric properties of the square $ABCD$. In a square, all sides are equal, and all angles are right angles. Suppose the side length of the square is $s$, so:

$AB = BC = CD = DA = s$ Also, since it is a square, the diagonals are equal and bisect each other at $45^\circ$.

Step 2: Placement of Points $M$ and $N$

• Let $M$ be a point on segment $AD$, and let the distance from $A$ to $M$ be $x$, so that $AM = x$ and $MD = s - x$.
• Let $N$ be a point on segment $DC$, and let the distance from $D$ to $N$ be $y$, so that $DN = y$ and $NC = s - y$.

Step 3: Angles at $M$

We are given two angles:

• $B\hat{M}A = 60^\circ$
• $N\hat{M}D = 60^\circ$

This means that the configuration around $M$ forms two equilateral triangles involving points $B, A, M$ and $N, D, M$.

Step 4: Use of Symmetry

The angles $B\hat{M}A$ and $N\hat{M}D$ being both $60^\circ$ suggests symmetry in the figure. Since these angles indicate equilateral triangle-like structures, and $M$ is on segment $AD$ while $N$ is on segment $DC$, it follows that the overall shape will exhibit rotational symmetry about point $M$.

Step 5: Calculating $M\hat{B}N$

The key observation is that because of the symmetry and the $60^\circ$ angles at $M$, the angle $M\hat{B}N$ is part of a cyclic quadrilateral formed by $M, B, N$. In such a configuration, the external angle (formed by extending $BM$ and $MN$) will sum to $360^\circ$ with the interior angles at $M$ and $N$.

Therefore, since both $B\hat{M}A$ and $N\hat{M}D$ are $60^\circ$, the angle $M\hat{B}N$ will be:

$M\hat{B}N = 180^\circ - (60^\circ + 60^\circ) = 60^\circ$

Thus, the required angle $M\hat{B}N = 60^\circ$.

Final Answer:

$M\hat{B}N = 60^\circ$

Would you like further details or have any questions?

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Here are 5 related questions to deepen understanding:

1. How can we generalize the result if $B\hat{M}A = N\hat{M}D = \theta$ instead of $60^\circ$?
2. What is the significance of symmetry in problems involving squares and specific angle conditions?
3. How would the problem change if the figure was a rectangle instead of a square?
4. What geometric properties of squares make angle-chasing a useful technique?
5. Can this problem be solved using coordinate geometry or vectors for verification?

Tip: In problems involving angles and symmetry, look for possible rotational or reflective symmetries to simplify the calculation.