Find the sum of angles without trigonometry? I have found the sum it's $180$ but using right triangle and sine theorem.


Draw a right triangle $ABC$ with the following properties:
$A$ is at the origin.
$C$ is the right-angle vertex at $(1,1)$.
$B$ is on your "left" as seen along a line of sight from $A$ through $C$.
$BC$ is twice as long as $AC$.
Then $B$ lies at $(-1,3)$ and the straight angle at the origin, in the upper half plane, is partitioned into $\arctan(1)+\arctan(2)+\arctan(3)$.
