A plane mirror is inclined at angle `theta` with the horizon. If a ray incident from 10 degree above the horizon becomes vertical after reflection from this mirror, what is `theta?`

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Please look at the picture attached. The horizon is drawn in blue, a mirror in green, an incident ray and its continuation in red and a reflected ray and its continuation in orange.

Denote the given angle between an incident ray and horizon as `alpha` and the angle between an...

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Please look at the picture attached. The horizon is drawn in blue, a mirror in green, an incident ray and its continuation in red and a reflected ray and its continuation in orange.

Denote the given angle between an incident ray and horizon as `alpha` and the angle between an incident ray and a mirror as `beta.` The law of reflection when applied to a plane mirror implies that the angle between a reflected ray and a mirror is the same.

Also, two more angles with the vertex at the reflection point `O` are equal to `beta` as vertical angles. These four angles are marked as `beta` on the picture.

From the left right triangle `ACO,` we see that `beta=90-theta.` From the other hand, an angle `beta` of inclination is an external angle for the triangle `AOB` with the opposite angles `alpha` and `theta,` so `beta=alpha+theta.`  From these two equations we obtain `90-theta=alpha+theta,`  so `theta=45-alpha/2=45-5=40` (degrees). Therefore `beta=90-theta=50` degrees. This is the answer.

Note that an incident ray may go from below and become vertical but downwards. In that case, the picture, the solution and the answer will be the same.

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