I'm looking for a way to look at a triangle, and perhaps visualize a few extra lines, and be able to *see* that the interior angles sum to $180^\circ$.

I can visualize that supplementary angles sum to $180^\circ$. I'd like to be able to *see* the interior angle sum similarly...

I can see that the exterior angles must sum to $360^\circ$, because if you walked around the perimeter, you would turn around exactly once (though I can tell this is true, I don't really see it). I also saw a proof on KA, where the exterior angles were superimposed, to show they summed to $360^{\circ}$ (though I'm not 100% comfortable with this one).

Finally, for $a$, $b$, and $c$ exterior angles $a+b+c=360$:

\begin{align} (180-a) + (180-b) + (180-c) & = 3\times 180 - (a+b+c) \\ & = 3\times 180 - 360 \\ & = 180 \\ \end{align}

But I find this algebra hard to *see* visually/geometrically. Is there proof that enables one to directly *see* that the interior angles of triangle sum to $180^\circ$?

A couple of secondary questions:

- am I visually deficient in my ability to imagine?
- or, am I asking too much of a proof, that I be able to
*see*it, and that beimg able to tell that it is true should be enough...?