4.52. The solid angle at the $i$-th vertex of a polyhedron is $\sigma_{i}-\left(n_{i}-2\right) \pi$, where $\sigma_{i}$ is the sum of the dihedral angles along the edges emanating from it, and $n_{i}$ is the number of these edges (see problem 4.44). Since each edge emanates from exactly two vertices, $\sum n_{i}=2 \mathrm{P}$, where $\mathrm{P}$ is the number of edges. Therefore, the sum of the solid angles of the polyhedral angles is $2 \sigma-2(\mathrm{P}-\mathrm{B}) \pi$, where $\sigma$ is the sum of the dihedral angles, and $\mathrm{B}$ is the number of vertices. It remains to note that $\mathrm{P}-\mathrm{B}=\Gamma-2$ (problem 8.14).

V. V. Prasolov, I. F. Sharygin