Proving theorems and special cases (Part 10): Angles in a convex n-gon

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve seen that a conjecture could be true for the first 40 cases or even the first cases yet not always be true. We’ve also explored the computational evidence for various unsolved problems in mathematics, noting that even this very strong computational evidence, by itself, does not provide a proof for all possible cases.

However, there are plenty of examples in mathematics where it is possible to prove a theorem by first proving a special case of the theorem. For the remainder of this series, I’d like to list, in no particular order, some common theorems used in secondary mathematics which are typically proved by first proving a special case.

1. Theorem. The sum of the angles in a convex n-gon is degrees.

This theorem is typically proven after first proving the following lemma:

Lemma. The sum of the angles in a triangle is degrees.

Clearly the lemma is a special case of the main theorem: for a triangle, and so . The proof of the lemma uses alternate interior angles and the convention that the angle of a straight line is 180 degrees.

Using this, the main theorem follows by using diagonals to divide a convex n-gon into triangles. (For example, drawing a diagonal divides a quadrilateral into two triangles.) The sum of the angles of the n-gon must equal the sum of the angles of the triangles.

So it is possible to prove a theorem by proving a special case of the theorem. Using the sum of the angles of a triangle to prove the formula for the sum of the angles of a convex n-gon is qualitatively different than the previous computational examples seen earlier in this series.