We all know that the sum of the interior angles of a polygon is $180^{\circ} (n-2)$. But is the converse true? Given a sequence of $n$ angle measures whose sum is $180^{\circ} (n-2)$, can it be realized by a polygon?
Let me be more precise. Let $n \ge 3$ be a positive integer, and let $\alpha_1, \ldots, \alpha_n$ be a sequence of real numbers so that $0 < \alpha_i < 360$ for all $i$, and $$\sum_{i=1}^n \alpha_i = 180(n - 2).$$
Does there necessarily exist a non-intersecting plane polygon $P_1P_2\ldots P_n$ so that the measure of $\angle P_{i-1} P_i P_{i+1}$ is $\alpha_i$ degrees for all $i$, assuming that $P_0 = P_n$?