In case of a perfectly in-elastic collision, the momentum is conserved and the light chooses the path which takes the least time to reach the final point which creates a path and results in reflected angle being equal to incident angle.

Angle of Incidence and Angle of Reflection

The relationship between the angle of incidence and the angle of reflection is a fundamental concept in the study of optics and reflection. This relationship is known as the law of reflection. Mathematically, this law can be expressed as follows:

Angle of Incidence (θ_i) = Angle of Reflection (θ_r)

To understand and prove this mathematically, let's break it down into the following sections:

1. Definitions:
- Angle of Incidence (θ_i): The angle between the incident ray and the normal (perpendicular line) to the reflecting surface at the point of incidence.
- Angle of Reflection (θ_r): The angle between the reflected ray and the normal to the reflecting surface at the point of reflection.

2. Geometry of Reflection:
- When a ray of light strikes a reflecting surface, it undergoes reflection, which involves two processes: incidence and reflection.
- The incident ray is the incoming ray of light, and the reflected ray is the ray that bounces off the reflecting surface.
- The normal to the reflecting surface is a line perpendicular to the surface at the point of incidence or reflection.
- The angle of incidence is measured between the incident ray and the normal, while the angle of reflection is measured between the reflected ray and the normal.

3. Law of Reflection:
- The law of reflection states that the angle of incidence is equal to the angle of reflection.
- This law holds true for any reflecting surface, assuming the surface is smooth and the medium remains the same.

4. Mathematical Proof:
- Let's consider a ray of light incident on a reflecting surface, forming an angle of incidence (θ_i) with the normal.
- By the law of reflection, the reflected ray will form an angle of reflection (θ_r) with the normal.
- To prove that θ_i = θ_r mathematically, we can use the concept of alternate interior angles.
- Consider the incident ray and the reflected ray as two lines intersected by the normal.
- The angle of incidence (θ_i) and the angle of reflection (θ_r) are alternate interior angles formed by these intersecting lines.
- According to the properties of alternate interior angles, when two parallel lines are intersected by a transversal, the alternate interior angles are congruent.
- Therefore, θ_i = θ_r, proving that the angle of incidence is equal to the angle of reflection.

In conclusion, mathematically proving that the angle of incidence is equal to the angle of reflection involves understanding the definitions, geometry of reflection, and applying the law of reflection. By considering the concept of alternate interior angles, the equality of these angles is established.

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