# Proof that angle of incidence is equal to angle of reflection, geometrically

Angle of incidence = Angle of reflection is true experimentally.
I just started highschool. Don't mind if this is a low level question. But I was just curious.

• This can be proved by Huygens Principle, and you must be familiar with waves, wavefront, and similar concepts. Commented Nov 5, 2021 at 17:30
• I am not familiar with Huygens principle now Commented Nov 6, 2021 at 6:00

A proof requires axioms, basic statements that can't be proven. In the case of physical principles, some of these will be just reliable observations.

Fermat's Least Time Principle tells us that light take the least time path between possible trajectories. If the speed of light remains constant in the material, the principle of Least Time becomes a principle of least distance. These will serve to guide a geometric argument for equal angles.

Suppose you have two points A and B above a mirror at different heights above the mirror. Find a point M on the mirror between these two such that the sum of the distance from A to M and from M to B is minimized.

One can use the Pythagorean theorem and calculus to solve this. One could also consider the problem if a point B' is added as a reflection of the point B. The shortest distance between two points is a straight line and by the Pythagorean Theorem it can be shown that the distance form M to B and the distance from M to B' are the same for any point M. So the line from A to B' intersects the mirror at a point M that minimizes A to M to B travel distance/time.

The argument is reversible starting from B and traveling to a reflected point of A, call it A'. This line also intersects the mirror at M.

Now consider a line dropped from B to B' and call where it intersects the mirror N. Let P be a point also on the mirror on the opposite of M from N but the same distance. It can be proven triangles MNB and MNB' are congruent by the SAS rule of congruent triangles. Angle AMO and NMB' are vertical angles and vertical angles are congruent. The angle of incidence of ray is measured from the normal to the plane it intersects and congruent angles have congruent angles have congruent complements, so it follows that the angles of incidence are congruent.

• I appreciate your sincere efforts Commented Nov 6, 2021 at 6:01

There are no mathematical proofs of physical phenomena. The universe is a real thing that is what it is; mathematics is a logic construct with no physical reality.

Theories explain the universe, often in terms of models. A model is a mathematical approximation of a system that we can manipulate with logic to predict the behavior of the system. The universe has internally consistent logic, so mathematical models can predict its behavior.

Proofs can extend the implications of a model, which may turn out to let us predict the behavior of the universe in a new way, confirm the validity of a model by showing that it predicts well-known phenomena, or may merely point out an imperfection in the model. Ultimately experimentation must decide which.

• I appreciate your sincere efforts Commented Nov 6, 2021 at 6:01