1. If alternate interior angles are congruent, then the two lines are parallel.
2. If two lines are parallel, then alternate interior angles are congruent.We discussed the proofs as laid out in the textbook (we both — the student in San Antonio and I here in Houston — could look at the same material this way), going over what they said, how the proofs were laid out, and why the proofs were laid out the way they were. The student got a good, solid presentation of the proofs — not the kind of thing that is done very much in modern education. First off, the very important proofs of geometry are usually neglected or devalued/deemphasized today, which results in students who do not appreciate — and who do not know how to engage in the process of — reasoning. And second, the very important “hows” and “whys” of proofs are even more infrequently taught. I focus on the “how” and the “why” of proofs, as well as the “what.” Geometry should be used to help raise a student’s thinking from the oral and semi-literate to the literate; it should be used to raise a student’s thinking from the subconscious and perceptual to the volitional and conceptual. I milk theorems and proofs for their methods and logic, as well as their content. Cannot have the former without the latter. But I pointed out how the first proof depended on the basic idea (which we learned prior to this) that an exterior angle of a triangle is greater than either remote interior angle, while the second proof depended on the idea (which we learned prior to this) that you can have only one parallel to a given line through a point not on the line. As this illustrates, my students learn hierarchy (the structure of knowledge), an important principle of logic. A fascinating thing about these two theorems is a new method used: reductio ad absurdum. My students learn that reductio is a form of argument very important in philosophy, as well as law and mathematics. They learn that it has played an important role in physics, too — which role my students learn about in detail. I concretize reductio; I do not leave students floating or frustrated intellectually with empty promises or empty leads. As this illustrates, my students learn integration (the interrelationship of all knowledge), an important principle of logic. And they learn to concretize their abstractions, to not make statements not backed up by facts. I take logic seriously, so I take hierarchy and integration seriously — I walk the walk and talk the talk. It takes someone well-versed in geometry — and well-versed beyond — to make these points. There is no taking the place of a really good teacher. The reductio arguments in these two proofs would go by the student; very rare would be the student who grasped the importance of reductio — students (and most adults) just don’t have the context. Invaluable is the teacher who stops the student, points out the nature of the argument, names it reductio ad absurdum, identifies the importance of the argument, and gives examples of where the argument has been used in history. It is critical to show the student how knowledge of reductio helps him/her (1) in reasoning and (2) in life. And this is the role only the master teacher can fill. After going over those two theorems, we discussed some corollaries of them — while discussing what a corollary was. And at this point we were able to prove — finally — that the angles of a triangle add to 180! I like this proof. Yeah, it’s not as interesting or complex as another favorite proof of mine: the geometric proof that y = x^2 in a parabola, but it’s a nice, solid, fundamental proof. Here’s a (poor-quality, semi-blurry — but, hey, what do you expect from a cell phone camera?…at this point in time) picture of my whiteboard notes (as seen on my computer screen!) showing the diagram and proof (well…a sketch of a proof): And as a corollary, we were able to prove that the exterior angle of a triangle is equal to the sum of the remote interior angles. And here’s a picture for the exterior angles theorem:
At this point, we were out of time, so I took a few questions (he wanted to review a proof of the Pythagorean Theorem, of which he has learned two so far) and gave the student his assignments for the week. All in all, an excellent class.
It is a joy — and quite a sense of accomplishment — to see students grow cognitively/intellectually, and to see them enthusiastic about it.
(c) 2009 Michael Gold