Problem 776

$\Delta$ If the sum of these five angles is a full circle, that is, two straight angles, which is $2 \times 180^{\circ}=360^{\circ}$. Each of them is one fifth of $360^{\circ}$, that is, $360 / 5=72^{\circ}$. $\triangleleft$

This problem and its solution require some comments. The reason is that different geometry textbooks interpret the concept of an angle differently. One can consider an angle as a geometric figure composed of two rays. Or one can consider an angle as the part of the plane contained between these rays. With this understanding, two rays with a common vertex define two angles: they are obtained by making cuts along the sides. One of these angles is less than a straight angle, and the other is greater.

With the first approach, the maximum possible angle is $180^{\circ}$ (a straight angle), while with the second approach, it is $360^{\circ}$ (a full circle, which can be conditionally considered as the supplement to an angle of $0^{\circ}$).

What does this have to do with the previous problem? Here's how: if we allow angles greater than $180^{\circ}$, everything is simple - an angle of $360^{\circ}$ is divided into 5 equal parts, the sum of which is $360^{\circ}$, and so on. If, however, we do not want to talk about angles greater than $180^{\circ}$, then this reasoning does not work and we need to act a bit more complicated. Draw any line through the given point, which will divide two of the five angles into parts. In total, there will be 7 parts. The sum of these parts is the same as the sum of our five angles (each cut angle is equal to the sum of its two parts). On the other hand, these parts add up to two straight angles, so in total they give $2 \times 180^{\circ}$.