Wednesday, July 30, 2014

A Preview: Images

Let's include a few images of the theorems mentioned in the previous post. But before I do so, let me make a few comments to clarify some of what I wrote in my last post.

I wanted to rewrite the proof that if a line is parallel to its reflecting line (its mirror), then its parallel to its image as well, so that it is a direct proof. Some may wonder whether this Line Parallel to Mirror Theorem is even worth giving in a geometry class. Going back to the Common Core Standards:

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

So Common Core asks students to know the conditions when a dilation maps a line to a parallel line. Also, Wu proves the conditions when a rotation (of 180 degrees) maps a line to a parallel line. And now I give the conditions when a reflection maps a line to a parallel line. So we see that it's indeed proper to give the Line Parallel to Mirror Theorem.

And in all three cases, the proofs are similar. Taking our cue from Wu, we first find the conditions under which the transformation maps a line to itself. Such a line, by the way, is often called an invariant line of that transformation. Then, to prove that a given line, which we'll call L following Wu, under the right conditions is parallel to its image L', we first choose a point P on L'. Then there must be a point Q on L such that Q' is P, and Q and P lie on an invariant line l of the transformation. Then by the Line Intersection Theorem, L and l intersect at exactly one point -- and that point is Q, not P. Since P is already on l, it can't lie on L. But P is any arbitrary point on L'. So L and L' can't have any points in common -- they are parallel.

Now here are the images:

Notice that this is still a work in progress. I'm not sure whether all of these would be appropriate for a high school class, based on the order in which I will present the material. In particular, the Perpendicular to Parallels Theorem is clearly a generalization of the Corresponding Angles Theorem, the difference being that the latter requires rotations while the former requires only reflections -- and I plan on presenting reflections before rotations. But I also plan on presenting rotations before Playfair's Postulate, so by the time we get to a Playfair proof, we might as well use the rotation version.

This ends the preview. I've decided that I will officially cover the material over the course of an academic year, the same way it would be done in the classroom.

But lately I've noticed that there are two school years commonly followed in this country. The traditional calendar has school start after Labor Day in September. This results in the first semester ending around late January or so. The early-start calendar starts in August, so that an entire semester can be covered before Christmas -- therefore avoiding having winter break separate the finals from most of the semester.

So far, I'll follow an early-start calendar. So I'll begin with Chapter 1 in August at some point.

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