A figure on a plane has exactly two axes of symmetry. Find the angle between these axes.

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Sequential execution of two symmetries with respect to axes, the angle between which is $\varphi$, is a rotation by an angle of $2 \varphi$.

## Solution

Let the angle between the axes of symmetry be $\varphi$. By performing the symmetries sequentially with respect to these axes, we obtain a rotation by an angle of $2 \varphi$, or a parallel translation by a vector perpendicular to the axes (if $\varphi=0$). In the second case, the figure will have an infinite number of axes of symmetry, obtained from the given axes by translations by the specified vector and its multiples, which contradicts the condition.

In the first case, when the figure is rotated by an angle of $2 \varphi$, it transitions into itself, and therefore, its axis of symmetry transitions into an axis of symmetry. Since there are only two axes of symmetry, and the first axis transitions into the second axis when rotated by $\varphi$, it will transition into itself when rotated by $2 \varphi$. This means that $2 \varphi=180^{\circ}$.

## Answer

$90^{\circ}$.