Find the sum of angles without trigonometry?

Question

I have found the sum it's $180$ but using right triangle and sine theorem.

Answer 1

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.

Answer 2

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

Answer 3

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 $$ \alpha + \beta + \gamma = \frac{\pi}{4} + \frac{3\pi}{4} = \pi. $$

Answer 4

@Jack D'Aurizio thanks for your solution and suggestions i found a little bit different solution from yours :)