jmc

We know that the sum of the degree measures of the interior angles of a polygon can be found using the formula $180(n-2)$ where $n$ is the total number of sides of the polygon. Since the polygons in this problem are regular, each interior angle measure can be found by substituting the appropriate $n$ into the formula $\frac{180(n-2)}{n}$. From this, we know that $\angle DEA$, an interior angle of a regular pentagon, has degree measure $\frac{180(5-2)}{5}=108^{\circ }$. We also have that $\angle FEA$, an interior angle of a regular hexagon, has degree measure $\frac{180(6-2)}{6}=120^{\circ }$.

Finally, we know that the angle measures of $\angle DEA$, $\angle FEA$, and $\angle DEF$ must sum to $360^{\circ }$, so $\angle DEF$ has an angle measure of $360-108-120=\boxed{132}$ degrees.