We are given a square $ABCD$ and an equilateral triangle $CDE$, and the goal is to find the value of $\angle BAE$.

Analysis:

• Since $ABCD$ is a square, all angles are $90^\circ$, and the sides $AB = BC = CD = DA$ are equal.
• Since $CDE$ is an equilateral triangle, all angles in triangle $CDE$ are $60^\circ$, and $CE = DE = CD$.

Steps to solve:

1. Recognize the angles:

   • The angle at $\angle CDE = 60^\circ$, since it is part of the equilateral triangle.
   • $\angle BAE$ is part of triangle $BAE$, and we need to find its value using geometric properties of the figure.

2. Use geometric symmetry:

   • We know that $ABCD$ is a square, so lines like $AE$ and $AB$ may have specific relationships due to the symmetry of the square and triangle $CDE$.
   • Since triangle $CDE$ is equilateral, line $AE$ forms a diagonal-like structure across the square and triangle.

By analyzing the setup geometrically, I will calculate $\angle BAE$.The value of $\angle BAE$ is approximately $45^\circ$.

This result comes from the symmetry of the square $ABCD$ and the positioning of point $E$ along the diagonal, which allows us to use basic trigonometry to estimate the angle.

Would you like any further details on this solution, or have any questions?

Here are five related questions for further exploration:

1. How does the equilateral triangle $CDE$ affect other angles in the square?
2. Can the length of the diagonal $AE$ be expressed in terms of the side of the square?
3. How would the solution change if $CDE$ were not equilateral?
4. What would $\angle BAE$ be if the square were inscribed in a circle?
5. How does the concept of symmetry help in solving this type of problem?

Tip: When dealing with squares and equilateral triangles, symmetry and simple trigonometric ratios often lead to quick solutions.