[Properties of Symmetries and Axes of Symmetry]

How many axes of symmetry can a heptagon have?
Answer: 0, 1 or 7. An axis of symmetry of a heptagon must pass through one of its vertices (the other vertices are divided into pairs of symmetric vertices). Suppose a heptagon has an axis of symmetry. Then it has three pairs of equal angles and three pairs of equal sides. The second axis of symmetry can be positioned in three essentially different (non-symmetric) ways. It is easy to see that in each of these three cases, all sides of the heptagon turn out to be equal and all angles are also equal.