The Parallel Postulate and an unfortunate Pedagogical Shortcut
the text goofs, big, and two freshmen are able to do what the book says cannot be done
I teach very little Geometry. It is my least favorite high school course*. But I am teaching Geometry this term. Two sections. Advanced freshmen, who took Algebra in the Fall.
I do lots of “reasoning” preparation before we get to points, lines, planes, postulates, and proof…
So here we are, in March, delayed start (delayed by choice), following our text (Jurgenson Brown Jurgenson) pretty closely, and the text goofs, big, and the kids have enough preparation that they do what the book implies cannot be done. Well, two of them do. But they’re 9th graders, right?
What the hell is the Parallel Postulate?
When I was in school, I thought it was “given a line and a point not on that line, there is exactly one line parallel to the given line passing through the given point.” According to our textbook, the postulate they offer is “given two parallel lines cut by a transversal, corresponding angles are congruent.” And then there’s Euclid. Strange guy. His version: ” If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.”
Turns out, you can take your pick. Postulate one of these, and the others, plus a host more, can be proven as theorems. Sum of the angles of a triangle = 180. And I emphasized this. We can postulate one, and then prove the others, and we can choose which to postulate (ok, we’ll go with the book. But in theory…) Oh, and so we postulate that corresponding angles formed by a tranversal cutting parallel lines are congruent, and we prove as consequences a bunch of related stuff, including the 180 degrees in a triangle.
Next day… I write the converse on the board, and ask them to prove it (If corresponding angles are congruent, lines must be parallel). I’m going to let them frustrate, just a bit, and then tell them, yes, this is a theorem, not a postulate, but we’re not proving things this tricky yet. But I got two surprises.
1a. A student suggested proof by contradiction. Took our postulate. Used a theorem (if lines are parallel, same side interior angles are supplementary). Assumed same side interiors are supplementary, and that the lines do cross. The contradiction comes from the sum of the angles in the resulting triangle. Nice.
1b. A student (another) suggested a different proof by contradiction. He let the corresponding angles be congruent, but the lines not parallel. And then he added a parallel line to the picture (the angle between them is where the contradiction appears). None of the t’s were crossed or i’s dotted, but the direction was good.
2. Why did I not consult the book before altering my lesson??? The book lies. It should say “The converse of our version of the parallel postulate is a theorem. We do not have the tools to prove this theorem yet; we will prove it later, when we learn the special kind of proof that is required” But it says something else. It says that we have another postulate.
Ouch. I spent weeks readying my students for working in an axiomatic system. The game is to postulate as little as possible, but we have to postulate some things. We’d even studied, a bit, Bolyai, Lobachevsky, and the parallel postulate.
(We also did an extended logic unit, where they proved and proved, and even proved by contradiction, which is how I got two proofs)