Alternate Interior Angles

This is a note from Heath's translation of the Elements about the theorem(s) I'll talk about below. It serves as a better introduction than anything I thought of writing:

With this proposition [27] begins the second section of the first Book. . . . [This] section leads up to the third, in which we pass to relations between the areas of triangles, parallelograms, and squares, the special feature being a new conception of equality of areas, equality not dependent on congruence. This whole subject requires the use of parallels. Consequently the second section beginning at 1.27 establishes the theory of parallels.

It's probably not too unfair to paraphrase the above as saying "we need this parallelism stuff to get to some much cooler stuff in the next section of the book."

And it reminds me that a lot of what we learn about mathematics in school is similarly transitional, connecting a much more sparse population of robust or "interesting" topics. Focusing exclusively on these transitional topics—this connective tissue—or on the entire subject as transitional, is what mathematician Edward Frenkel refers to as "painting the fence" in art class.

But of course few of these interesting real-world topics would be interesting without the connective tissue. So, while it is certainly worth our while to use the 20% to get students to plow through the "boring" 80%, pretending that everything is in the 20% (the Lake Wobegon effect of curriculum design, if you will) is a delusion resting on a gross ignorance of reality and of mathematics.


All Fired Up for Learning Now?

With that said, this 'transitional' proof is a little bit interesting in the way Euclid seems to break some rules in order to get to his conclusion:

Let the straight line \(\small\mathtt{EF}\) falling on the two straight lines \(\small\mathtt{AB}\), \(\small\mathtt{CD}\) make the alternate angles \(\small\mathtt {AEF}\), \(\small\mathtt{EFD}\) equal to one another; I say that \(\small\mathtt{AB}\) is parallel to \(\small\mathtt{CD}\). For, if not, \(\small\mathtt{AB}\), \(\small\mathtt{CD}\) when produced will meet either in the direction of \(\small\mathtt{B}\), \(\small\mathtt{D}\) or towards \(\small\mathtt{A}\), \(\small\mathtt{C}\).
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We can see that this will be another proof by contradiction. He supposes that he is wrong and that lines \(\small\mathtt{AB}\) and \(\small\mathtt{CD}\) are not parallel. Therefore, they will meet at some point.

What's weird is how, in the diagram, these lines are made to meet—by making each one of them two lines instead of one! But the only reason that would be a problem is if anything in the rest of the proof relies on this distortion to be true. Let's see:

Let them be produced and meet, in the direction of \(\small\mathtt{B}\), \(\small\mathtt{D}\), at \(\small\mathtt{G}\). Then, in the triangle \(\small\mathtt {GEF}\), the exterior angle \(\small\mathtt{AEF}\) is equal to the interior and opposite angle \(\small\mathtt{EFG}\): which is impossible. [Because Proposition 16 of the first Book tells us that the exterior angle is always greater than either of the interior angles.]

Therefore \(\small\mathtt{AB}\), \(\small\mathtt{CD}\) when produced will not meet in the direction of \(\small\mathtt{B}\), \(\small\mathtt{D}\). Similarly it can be proved that neither will they meet towards \(\small\mathtt{A}\), \(\small\mathtt{C}\).

The mention of \(\small\Delta\mathtt{GEF}\) seems like a mistake, given the distortion in the drawing, but this triangle would exist even without the distortion if lines \(\small \mathtt{AB}\) and \(\small\mathtt{CD}\) were not parallel, since the lines would have to meet at some point.