What should be the angle between two plane mirrors so that whatever be the angle of incidence, the incident ray and the reflected ray from the two mirrors be parallel to each other

Options

A. `60^(@)`
B. `90^(@)`
C. `120^(@)`
D. `175^(@)`

Explanation

To find the angle between two plane mirrors such that the incident ray and the reflected ray from the two mirrors are parallel to each other, we can follow these steps:

### Step 1: Define the Angles
Let the angle between the two mirrors be denoted as θ. When a ray of light strikes the first mirror, it makes an angle of incidence, which we will denote as θ₁.

### Step 2: Reflection from the First Mirror
According to the law of reflection, the angle of reflection is equal to the angle of incidence. Therefore, the angle of reflection from the first mirror will also be θ₁.

### Step 3: Incident Ray on the Second Mirror
When the reflected ray from the first mirror strikes the second mirror, it will also make an angle θ₂ with the normal to the second mirror. Since the two mirrors are at an angle θ to each other, we can relate θ₁ and θ₂.

### Step 4: Relationship Between Angles
For the reflected ray from the first mirror and the incident ray on the second mirror to be parallel, the sum of the angles must equal 180 degrees. Thus, we can write:

\[ 2θ₁ + 2θ₂ = 180° \]

From this, we can simplify to:

\[ θ₁ + θ₂ = 90° \]

### Step 5: Expressing θ₂ in Terms of θ₁
From the equation \( θ₁ + θ₂ = 90° \), we can express \( θ₂ \) as:

\[ θ₂ = 90° - θ₁ \]

### Step 6: Analyzing the Geometry
Now, we need to consider the geometry of the situation. The angle between the two mirrors (θ) can be expressed in terms of θ₁ and θ₂. The relationship can be derived from the triangle formed by the angles:

\[ θ₁ + θ + θ₂ = 180° \]

Substituting \( θ₂ = 90° - θ₁ \) into this equation gives:

\[ θ₁ + θ + (90° - θ₁) = 180° \]

### Step 7: Simplifying the Equation
This simplifies to:

\[ θ + 90° = 180° \]

From which we can conclude:

\[ θ = 90° \]

### Conclusion
Thus, the angle between the two plane mirrors should be **90 degrees** for the incident ray and the reflected ray from the two mirrors to be parallel to each other.

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