BG-MO 1963 Problem 4

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that:

(a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal. (b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.