Shapovalov A.V.

Among the angles of each lateral face of a pentagonal prism, there is an angle $\varphi$. Find all possible values of $\varphi$.
In a right prism, each lateral face is a rectangle, so $\varphi=90^{\circ}$ is suitable.

Suppose $\varphi \neq 90^{\circ}$. We can assume that $\varphi<90^{\circ}$ (if the obtuse angles in the parallelograms are equal, then the acute angles are also equal). Draw in the plane of the base through one of the vertices $V$ lines parallel to each of the base edges. These lines form an angle $\varphi$ with the lateral edge $r$ emanating from $V$. However, there are no more than two such lines in this plane (all possible lines passing through $V$ at an angle $\varphi$ to $r$ form a conical surface; it intersects the base plane in no more than two lines). Thus, there are only two directions for the base edges. Therefore, no fewer than three edges are directed the same way. Two of them will turn out to be neighbors. Contradiction.

## Answer

$90^{\circ}$.