A ray of light incident on a plane mirror at an angle `theta`. If the angle between the incident and reflected rays is `80^@`, what is the value of `theta`?

Options

A. `40^@`
B. `50^@`
C. `45^@`
D. `55^@`

Answer

To solve the problem, we need to determine the value of the angle \( \theta \) given that the angle between the incident ray and the reflected ray is \( 80^\circ \).

### Step-by-Step Solution:

1. **Understanding the Angles**:
- Let \( i \) be the angle of incidence.
- Let \( r \) be the angle of reflection.
- According to the law of reflection, the angle of incidence \( i \) is equal to the angle of reflection \( r \). Thus, \( i = r \).

2. **Relation Between Angles**:
- The angle between the incident ray and the reflected ray is given as \( 80^\circ \).
- Therefore, we can express this relationship as:
\[ i + r = 80^\circ \]
- Since \( i = r \), we can substitute \( r \) with \( i \):
\[ i + i = 80^\circ \]
- This simplifies to:
\[ 2i = 80^\circ \]

3. **Calculating the Angle of Incidence**:
- Now, we can solve for \( i \):
\[ i = \frac{80^\circ}{2} = 40^\circ \]

4. **Using the Right Angle Property**:
- The angle \( \theta \) is the angle between the incident ray and the normal (the line perpendicular to the mirror surface).
- In a right-angled situation, we know:
\[ i + \theta = 90^\circ \]
- Substituting the value of \( i \):
\[ 40^\circ + \theta = 90^\circ \]

5. **Solving for \( \theta \)**:
- Rearranging gives:
\[ \theta = 90^\circ - 40^\circ = 50^\circ \]

### Final Answer:
The value of \( \theta \) is \( 50^\circ \).

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