In the diagram (which you can click or tap to rotate), we have \(\small\Delta\mathtt{ABC}\) with \(\overline{\small\mathtt{BC}}\) extended to point \(\small\mathtt{D}\). Also, \(\overline{\small\mathtt{CE}}\) is drawn so that it is parallel to \(\overline{\small\mathtt{AB}}\) (which the construction just before this proposition shows us how to do), and I've extended these parallel segments and also \(\overline{\small\mathtt{AC}}\) with dashed lines to help me visualize what's coming next.
Hover over the diagram to see the angle congruences. \(\small\measuredangle\mathtt{A}\) and \(\small\measuredangle\mathtt{ACE}\) are congruent because they are alternate interior angles—the converse of the theorem I wrote about here is used to justify this statement. And \(\small \measuredangle\mathtt{B}\) and \(\small\measuredangle\mathtt{ECD}\) are congruent because they are corresponding angles, which is also covered in that theorem (I.29).
Since pink + green = pink + green, the measure of the exterior angle, \(\small\measuredangle\mathtt{ACD}\), is equal to the sum of the opposite interior angles: \(\small \measuredangle\mathtt{A}\) + \(\small\measuredangle\mathtt{B}\). Finally, I.13 tells us that pink + green + m\(\small \measuredangle\mathtt{ACB}\) is equal to 180\(^\circ\). So, the sum of the measures of the angles of a triangle (in Euclidean geometry) is 180 degrees. Or, as Euclid might have said it, equal to two right angles.
Just a Couple of Notes
I've mentioned before that the notes given alongside these proofs can be more interesting than the proofs themselves. Here's one (a quote from Geminus) that can help to dispel the notion that geometry—and certainly mathematics in general—does not involve iteration and a gradual climb to truth:
It's also interesting for me to reflect on how much this proof acts like a real proof for me and I think other people. That is, we do some fairly deep reasoning to establish the general result—i.e., prove a theorem, or at least convince ourselves that the theorem is proved—and then we close the door. We are satisfied in the future to use the implications of this proof without revisiting the arguments all over again. Our re-entry point back into using the theorem is often breathtakingly superficial: something is or looks like a triangle. I'm not sure that a lot of other proofs (or other established concepts in mathematics) have this characteristic for those of us in K12 land.
Image-mask credit: Jimmie
