Problem 664

34. In an orthogonal trihedral angle, the three dihedral angles are right angles, and the sum of their measures is $\frac{3 \pi}{2}$. Subtracting $\pi$ from this, we get $\frac{\pi}{2}$ - the measure of the orthogonal trihedral angle.

Three mutually perpendicular planes divide space into 8 orthogonal trihedral angles. The sum of their measures is $8 \cdot \frac{\pi}{2}=4 \pi$. From additivity, it follows that this will also be the sum of the measures of any polyhedral angles with a common vertex, filling all space.