1. **Problem Statement:** We have a regular hexagon ABEFGH and a regular quadrilateral BCDE sharing the edge BE. We need to find the measure of angle $\angle DEF$. 2. **Key Properties:** - A regular hexagon has interior angles of $120^\circ$. - A regular quadrilateral (square) has interior angles of $90^\circ$. - Since BE is a common side, points B and E are shared. 3. **Analyze the angle $\angle DEF$:** - Point E is a vertex of both polygons. - In the quadrilateral BCDE, angle at E is $90^\circ$. - In the hexagon ABEFGH, angle at E is $120^\circ$. 4. **Determine the angle $\angle DEF$:** - The angle $\angle DEF$ is formed by points D, E, and F. - D and C are vertices of the quadrilateral, F is a vertex of the hexagon. - Since the polygons share edge BE, the angle between edges ED and EF at E is the sum of the interior angles of the two polygons minus $360^\circ$ (full circle) to find the external angle. 5. **Calculate $\angle DEF$:** - The interior angle at E in the hexagon is $120^\circ$. - The interior angle at E in the quadrilateral is $90^\circ$. - The angle between ED and EF is $360^\circ - (120^\circ + 90^\circ) = 360^\circ - 210^\circ = 150^\circ$. **Final answer:** $\boxed{150^\circ}$