Notes
two triangles in a square solution

Solution to the Two Triangles in a Square Puzzle

Solution by Symmetries of a Square, Angles in a Triangle, and Angles at a Point on a Straight Line

Label the points as above. Reflecting the square in the diagonal $B F$ swaps $A$ and $D$ and leaves $E$ where it is. Therefore, angles $B \hat{A} E$ and $E \hat{D} B$ are equal. Let us write this angle as $a^\circ$.

Then as angles in a triangle add up to $180^\circ$, and angles at a point on a straight line also add up to $180^\circ$, angle $A \hat{C} B$ is equal to $2 a^\circ$.

As triangle $A B C$ is right angled, angles $A \hat{C} B$ and $B \hat{A} C$ add up to $90^\circ$, so $3 a^\circ = 90^\circ$ meaning that $a^\circ = 30^\circ$. Therefore, angle $B \hat{A} C = 30^\circ$.