To solve for the degree measure of the exterior angle $\angle DEF$, we need to understand the geometric properties of the regular pentagon $ABCDE$ and the regular hexagon $AEFGHI$.

1. Interior Angles:

   • A regular pentagon has an interior angle of $\frac{(5-2) \times 180^\circ}{5} = 108^\circ$.
   • A regular hexagon has an interior angle of $\frac{(6-2) \times 180^\circ}{6} = 120^\circ$.

2. Exterior Angles:

   • The exterior angle of any polygon is $180^\circ - \text{interior angle}$.
   • For the pentagon $ABCDE$, the exterior angle is $180^\circ - 108^\circ = 72^\circ$.
   • For the hexagon $AEFGHI$, the exterior angle is $180^\circ - 120^\circ = 60^\circ$.

3. Analyzing $\angle DEF$:

   • $\angle DEF$ is formed at vertex $E$, which is shared between the pentagon and hexagon.

Considering the geometry:

• Since $E$ is a vertex of both polygons, the angles at $E$ add up to form a straight line, i.e., $180^\circ$.

• The angles at $E$ within each polygon are interior angles of those polygons: $\angle DEA = \text{interior angle of pentagon} = 108^\circ$ $\angle AEF = \text{interior angle of hexagon} = 120^\circ$

Since both angles sum to exceed $180^\circ$: $\angle DEF = 180^\circ - (\angle DEA + \angle AEF - 180^\circ)$ $\angle DEF = 180^\circ - (108^\circ + 120^\circ - 180^\circ)$ $\angle DEF = 180^\circ - 48^\circ$ $\angle DEF = 132^\circ$

Thus, the degree measure of the exterior angle $\angle DEF$ is $\boxed{132^\circ}$.

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

Here are 5 questions you can ask:

1. What are the properties of a regular polygon?
2. How do you calculate the interior angle of any regular polygon?
3. How do you find the sum of interior angles of a polygon?
4. Can you explain the relationship between interior and exterior angles of polygons?
5. How can you apply these angle properties to solve other geometric problems?

Tip: Always verify if multiple polygons share common vertices to utilize angle properties effectively.