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

 A: Just rearrange them and notice that the bold triangle is right and isosceles:

Another proof of $\arctan 1=\arctan\frac{1}{2}+\arctan\frac{1}{3}$ comes from:
$$ (3+i)(2+i) = 5+5i $$
by switching to arguments.
A: Consider following triangle:

As $BC = AC$ we have $\angle ABC = \angle CAB$ or
$$
\pi - \gamma - \beta = \gamma - \frac{\pi}{2} + \beta \iff \gamma + \beta = \frac{3\pi}{4}
$$
(here $\gamma$ is red angle from picture in question and $\beta$ is yellow one). It's obvious that green angle from question (detote it as $\alpha$) is equal to $\frac{\pi}{4}$. Thus we have
$$
\alpha + \beta + \gamma = \frac{\pi}{4} + \frac{3\pi}{4} = \pi.
$$
A: @Jack D'Aurizio thanks for your solution and suggestions i found a little bit different solution from yours :)

A: 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 $\arctan(1)+\arctan(2)+\arctan(3)$.
