In the diagram, $A B C D$ is a square, $\triangle A B E$ is equilateral, and $\triangle A E F$ is equilateral. What is the measure of $\angle D A F$ ?

![](https://cdn.mathpix.com/cropped/2024_04_30_9fccce2562493283ec28g-2.jpg?height=323&width=353&top_left_y=1259&top_left_x=1214)
We note that $\angle D A F=\angle F A B-\angle D A B$ and that $\angle F A B=\angle F A E+\angle E A B$.

Since $A B C D$ is a square, then $\angle D A B=90^{\circ}$.

Since each of $\triangle A E F$ and $\triangle A B E$ is equilateral, then $\angle F A E=\angle E A B=60^{\circ}$.

Therefore, $\angle D A F=\left(60^{\circ}+60^{\circ}\right)-90^{\circ}=30^{\circ}$.

ANSWER: $30^{\circ}$