If $P_{1} P_{2} \ldots P_{100}$ is a regular 100 -gon, what is the measure of the angle $\angle P_{20} P_{2} P_{1}$ in degrees?
Introduce the circumcircle of the polygon. The aforementioned angle cuts out an arc which is $\frac{81}{100}$ of that of the entire circle, i.e. which measures $\frac{81}{100} \cdot 360^{\circ}$. By the inscribed angle theorem, this means an answer of $\frac{81}{100} \cdot 180^{\circ}=145.8^{\circ}$.