Forum Replies Created
-
Posts
-
Bill,
I expect that you mean ‘very informal proofs’. But your description of how to go about it is right on. Of course, the diagram you describe is just the one used to prove SSS – join along a pair of sides of equal measure, construct a diagonal, and then invoke the Isosceles Triangle Theorem. As you say, the best way to prove the ITT is to construct the angle bisector and then ask what happens when we fold over it. Students get that right away.
Let me add that I find Harold Jacobs text Geometry: Seeing, Doing, Understanding just about right for a high school classroom. The one real deficiency from the point of view of the CCSS is that it makes SAS and a postulate. That’s a problem, though, that’s relatively easily rectified. (He also gives Euclid’s proof of the AA triangle similarity principle. By my lights, that’s fine. But one might like to do it instead with some sort of scale transformation postulate.)
I’ll respond to the question about parallels.
There’s no one right answer about what to call a postulate and what to call a theorem. Euclid offered a proof of the SAS triangle congruence principle in his Elements, but Hilbert, in his Foundations of Geometry, made that principle is postulate. Neither is right. Neither is wrong. They simply made different decisions.
But there are certain principles that should guide the choice of a set of postulates. One is that it should not include superfluous postulates, i.e. postulates that can be proven on the basis of other postulates already in place. I do think that a bit of that is fine in a high school geometry classroom, but it should be kept to a minimum. That’s why I so dislike, for instance, when a text makes both SAS and SSS triangle congruence postulates. SAS can be used in a relatively straightforward proof of SSS!
This principle implies that we should not make the statement below (or any other equivalent to it) a postulate:
When a transversal cuts a pair of lines so that alternate interior angles are congruent, then those lines are parallel.
Why not? It’s provable! See Book I, Proposition 27 of the Elements.
A second principle that should guide the choice of a postulate set is that the postulates chosen should be both simple and obviously true. (I mean this to hold only for the high school classroom. These requirements are dropped at higher levels.) That’s why, when I teach parallels, I choose the Playfair Postulate. (Through a point not on a line, there’s at most one line parallel to that given line.) It’s clear and (to students’ minds) obviously true. Together with the proposition above, it can be used to prove that if a pair of parallel lines are cut by a transversal, then alternate interior angles are congruent. (Here’s a quick sketch of the proof. Assume that point P does not lie on line m. Construct line n parallel to m through P. Construct a transversal to m and n through P. If alternate interior angles are not congruent, then we can construct a second line r through P for which they are. But then this line r will be parallel to m, and so we then have two lines through P parallel to m. This contradicts the Playfair Postulate. Hence alternate interior angles are in fact congruent.)
In my class I plan to keep the examples simple. This means rotations that are multiples of 90 degrees. Other rotations are more difficult to handle. How does one demonstrate, for instance, that one figure is the rotation of another 30 degrees about the origin? In general, that isn’t trivial, and it is of course beyond an introductory geometry class.
For my own part, I don’t think that analytic verification of a given transformation is of such great importance. A bit is good. But I do only so much as is necessary to motivate a set of results that follow: when polygons are congruent, sides which correspond and angles which correspond have equal measures; when sides and angles of polygons can be paired up in such a way that those which correspond have the same measure, then the polygons are congruent; SAS congruence and the rest of the triangle congruence principles.
Wu’s article is superb. It’s been my primary source as I rework my geometry class.
I take it that the core definitions – of ‘rigid motion’, say – can be given either in terms of transformation of the plane or transformation in the plane. In the first, we map the whole of the plane to itself. In the second, we map one subset of the plane to another. I’m tempted to say that it’s just a matter of taste.
