Consider two semi–infinite plane mirrors with their finite ends placed together. The angle subtended by the two mirrors is $\alpha$ . A ray, that is parallel to mirror 2, is incident on mirror 1 and after 8 reflections, it emerges parallel to mirror 1 (after a final reflection from mirror 2) Determine $\alpha$ . If  $\alpha =\frac{\pi }{n}$, Find $n$ = ?

Due to the symmetrical nature of the set-up and the reversibility of light rays, the path of the light ray between the  $4^{th}$ reflection (with mirror 2) and the $5^{th}$  reflection (with mirror 1) must be symmetrical about the symmetry axis of the two mirrors as well. That is, the path must be perpendicular to the symmetry axis. Then, the angle subtended by the ray after emerging from the $4^{th}$ reflection (with mirror 2) and mirror 2 must $\frac{\pi }{2}-\frac{\alpha }{2}$ (we are referring to the angle closer to the point of connection of the mirrors)

$4^{th}$ reflection is labeled above. We have 

$4\alpha +(\frac{\pi }{2}+\frac{\alpha }{2})=\pi \Rightarrow \alpha =\frac{\pi }{9}$