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…?
Best Answer
Since that fact about the angle sum is equivalent to the parallel postulate, any visualization is likely to include a pair of parallel lines. Here's one from wikipedia: