According to the above figure, $S_1$ and ${S^{\text{'}}}_1$ are the reflection of each other in $S_2.$

And $S_2$ and ${S^{\text{'}}}_2$ are the reflection of each other in $S_1.$

So, ${S^{\text{'}}}_1$ and ${S^{\text{'}}}_2$ are coincide.

$\Rightarrow 3\theta =\mathrm{\pi }$

$\Rightarrow \theta =\frac{\mathrm{\pi }}{3}$