Authors: Feldman G. Baranov D.V.

An angle is drawn, and only a compass is available.

a) What is the minimum number of circles that need to be drawn to definitely determine whether the given angle is acute?

b) How can one determine if the given angle is $31^{\circ}$ (any number of circles can be drawn)?
a) Let's draw a circle with its center on one side of the angle, passing through its vertex $O$. If it intersects the other side of the angle, the angle was acute.

b) Let's draw a circle with center $O$. Suppose it intersects the sides of the angle at points $A$ and $B$. Starting from point $A$, we will lay off chords equal to $AB$ on this circle. If by laying off 360 such chords, we circle the circumference exactly 31 times, then the angle $AOB$ is $31^{\circ}$.

## Answer

a) One.