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Regular pentagon $ABCDE$ and regular hexagon $AEFGHI$ are drawn on opposite sides of line segment $AE$ such that they are coplanar. What is the degree measure of exterior angle $DEF$?

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.

Discover how to find the degree measure of the exterior angle DEF formed between a regular pentagon ABCDE and a regular hexagon AEFHI sharing segment AE. This geometry problem involves understanding interior and exterior angles of polygons and their relationships. Suitable for advanced high school students.

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