Given soilid $\Gamma$ in three-dimensional Euclidean space $E^3$ has only two planes of symmetry. Prove that this planes are perpendicular.
I tried to show that composition of symmetries is also symmetry and if angle $a \neq \frac{\pi}{2}$ than composition is a rotation on angle $2a \neq \pi$. What else can be done to show contradiction?
Hint. If $\omega$ is the angle of the two planes $P$ and $Q$, i.e. $$ \omega=\widehat{PQ}, $$ then notice that every other plane $Q'$, which contains the intersection line of $P$ and $Q$, and for which $$ \widehat{PQ'}=n\omega $$ for some $n\in\mathbb n$, then $Q'$ is also a plane of symmetry.