1. **Problem statement:** Find the angle $\angle HFE$ in the given geometric figure. 2. **Identify known elements:** To find $\angle HFE$, we need information about points $H$, $F$, and $E$ and their positions or any given angles or side lengths related to these points. 3. **Formula and rules:** The angle between two lines or segments meeting at a point can be found using geometric properties such as the Law of Cosines, angle sum properties, or trigonometric ratios if coordinates or lengths are known. 4. **Intermediate work:** Since the problem does not provide additional data, we assume the points form a triangle or polygon where $\angle HFE$ is an interior angle. 5. **Explanation:** Without specific lengths or coordinates, the angle cannot be numerically calculated. If you provide side lengths or coordinates, we can use the Law of Cosines: $$\cos(\angle HFE) = \frac{FH^2 + FE^2 - HE^2}{2 \cdot FH \cdot FE}$$ where $FH$, $FE$, and $HE$ are the lengths of the sides opposite to the respective vertices. 6. **Conclusion:** Please provide the lengths of sides $FH$, $FE$, and $HE$ or coordinates of points $H$, $F$, and $E$ to calculate $\angle HFE$ precisely.