Two plane mirrors are inclined to one another at an angle of 60 degree

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As we can see from the figure, $\angle{M_2}AB$ is the corresponding angle of $\angle{M_2}O{M_1}$. Therefore, $\Rightarrow \angle{M_2}AB = \angle{M_2}O{M_1} = 60^\circ $Now, the line$A{N_2}$ represents the normal to the mirror ${M_2}$. Since the normal makes an angle $90^\circ $ with the plane of the mirror ${M_2}$, we can write\[\angle{M_2}A{N_2} = 90^\circ \]It implies,\[\angle{M_2}AB + \angle BA{N_2} = 90^\circ \\\Rightarrow 60^\circ + \angle BA{N_2} = 90^\circ \\\Rightarrow\angle BA{N_2} = 30^\circ \]$\angle BA{N_2}$ is the angle of reflection of the ray from the mirror ${M_2}$. Since the angle of reflection equals the angle of incidence on a plane mirror, the angle of incidence of the ray on the mirror ${M_2}$ is, $\angle{N_2}AD = 30^\circ $. Therefore, we get \[\angle{N_2}AO = 90^\circ \]It implies,\[\angle{N_2}AD + \angle DAO = 90^\circ \\\Rightarrow 30^\circ + \angle DAO = 90^\circ \\\Rightarrow\angle DAO = 60^\circ \]We know that in , the sum of all the angles is equal to $180^\circ $. Therefore,$\angle DAO + \angle AOD + \angle ODA = 180^\circ \\\Rightarrow 60^\circ + 60^\circ + \angle ODA = 180^\circ \\\Rightarrow\angle ODA = 60^\circ $We know that the angle of incidence of the ray on the mirror ${M_1}$ is $i$. Since the angle made by the normal ${N_2}D$ with the plane of the mirror is $90^\circ $, we can write$ \angle ODA + \angle AD{N_2} = 90^\circ \\\Rightarrow 60^\circ + i = 90^\circ \\\therefore i = 30^\circ $

Therefore, the angle of incidence $i$ is $30^\circ $.

Note: It is to be noted that the normal is a line drawn perpendicular to the surface of a mirror. The angles made by the reflected ray with the normal and the incident ray with the normal are always equal. Also, in questions of this type where it is given that the two mirrors are inclined at an angle $\theta $, and the incident ray on the first mirror reflects from the second mirror parallel to the first, the incident ray angle \[i\] is $2\theta - 90^\circ $. So, the incident angle can be found directly without doing many steps.

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Two plane mirrors are inclined to one another at an angle of 60 degree

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