Why is the sum of all external angles in a convex polygon $360^\circ$?
From my understanding, for each vertex in a convex polygon, there exist exactly $2$ exterior angles corresponding to it, which are both equal, vertically opposite, and add up to $180^\circ$ with the interior angle. If we take as true that sum of interior angles in a triangle is $(n-2)180^\circ$ degrees, then $$\sum_i 2\cdot (180^\circ-\alpha_i) = n\cdot 360^\circ - (n-2)\cdot 360^\circ = 720^\circ.$$ Am I missing something here?