Consider the diagonals of a 21-gon. Prove that at least one angle of less than 1 degree is formed.

Question

I think it should be solved using the pigeonhole principle. The answer is:

A $21-$gon has $189$ diagonals. If through a point in the plane, we draw parallels to these diagonals, $2 × 189 = 378$ adjacent angles are formed. The angles sum up to $360^\circ$, and thus one of them must be less than $1^\circ$.

I do not understand what is meant by "we draw parallels to these diagonals" nor where $360$ came from. The only solution I could think of was drawing all the diagonals through the same point and then counting how many angles "make up" the full circle of angles; I do not know if this is the correct interpretation.

Answer 1

The reasoning given is correct. By "we draw parallels to these diagonals," it means "we draw the unique line parallel to a given diagonal, for every single diagonal."

However, if $0-$degree angles are not counted as "angles", it is false.

Proof: Every diagonal of a regular $21$-gon is in one of at most $21$ directions which have angles of $8\frac{4}{7}$ degrees with each other. Therefore, the minimum angle in this case is at least $8\frac{4}{7}$ degrees, which is much greater than $1$ degree.

In fact, $8\frac{4}{7}$ degrees is optimal, since all directions of the diagonals that connect a vertex and another vertex $10$ edges away are different (since they either intersect at a vertex or inside the polygon, and therefore cannot be parallel), and $8\frac{4}{7}$ degrees is optimal for $21$ different directions.

Answer 2

A generalization of the Inscribed Angle Theorem states that the measure of a pair of vertical angles made by two chords is equal to half the sum of the measures of the two arcs subtended by those angles. (When the chords intersect on the circle, one of those arc-measures vanishes and we get the IAT. When extended chords meet outside the circle, we take half the difference of the arc measures.)

$$\theta=\frac12(\alpha+\beta)$$

When the endpoints of the chords are vertices of an $n$-gon, then those arc-measures are integer multiples of $360^\circ/n$, so that the angle measure is a non-negative integer multiple of $180^\circ/n$. If the chords are distinct and non-parallel, then the smallest possible measure is $180^\circ/n$ (attained when the chords meet at a common vertex and have adjacent vertices for their "other" endpoints; attained more generally by extended chords when the numbers of subtended edges differ by $1$). When $n=21$, this is $60^\circ/7$, which is greater than $1^\circ$.