Section 6-3: Rotations (Day 37)

We have entered the part of our plan that I am devoting to the writings of Dr. Hung-Hsi Wu. As I mentioned last week, I don't like jumping around the U of Chicago text so much, but it is impossible for me to cover the Wu curriculum without such skipping. Let me explain.

First, here's a link to the Wu text to which I am referring -- I've given it before, but I want to give it again because this is what I'm covering right now:

http://math.berkeley.edu/~wu/CCSS-Geometry_1.pdf

Now our focus is on the high school course which begins on page 73 at the above link, not the eighth grade course. But this is a good place to remind everyone that much of geometry taught in the early years, before the eighth grade, aren't that much different before and after Common Core. The eighth grade is when the focus on transformational geometry begins.

Now our goal is to prove Wu's Theorem 12:

"Thoerem [sic] 12. If a pair of alternate interior angles or a pair of corresponding angles
of a transversal with respect to two lines are equal, then the lines are parallel."

This theorem finally gets us into our usual Parallel Tests in terms of alternate interior, same-side interior, and corresponding angles. Now the proof of Theorem 12 depends on Theorem 1:

"Theorem 1. Let L be a line and O be a point not lying on L. Let R be the 180-degree
rotation around O. Then R maps L to a line parallel to L itself."

And Theorem 1 discusses rotations. Therefore, our focus right now will be on the concept of a rotation, our next major transformation after the reflection.

The U of Chicago text already contains a section on rotations -- but not until Section 6-3. (Once again, this refers to my own copy of the text, which is dated 1991. I've heard that newer versions of the text place rotations right in Chapter 4, along with reflections and translations.) Yet I plan on adding on material from Wu himself and other sources, since the U of Chicago doesn't prove anything like Wu's first theorem in Section 6-3, or anywhere else.

Notice that my order of the transformations (reflection-rotation-translation) differs from both that of the U of Chicago (reflection-translation-rotation) and that of Wu (rotation-reflection-translation). I must emphasize that one major problem with any reordering of chapters is circularity -- we may end up placing a theorem whose proof depends on another theorem prior to that theorem. Wu places rotations first because his proof that reflections are even well-defined depend on rotations. The U of Chicago text must put reflections first because it uses reflections to define rotations. This text ensures that reflections are themselves well-defined by including such in the Reflection Postulate.

Since I already started the U of Chicago order by giving reflections first, we must remind ourselves why I don't continue the U of Chicago order and give translations next. The problem is that the U of Chicago's definition of translation is not valid in non-Euclidean geometry and requires a Parallel Postulate to be valid -- so we end up with circularity if we try to derive the properties of parallel lines from translations. On the other hand, the definition of rotation is valid in non-Euclidean geometry, so we can give it without worrying about a Parallel Postulate.

So reflections and rotations, unlike translations, have the common property that they work in all types of geometry. Another common property is that reflections and rotations -- but not translations (unless it is the identity translation) -- have fixed points. The fixed point of a rotation is its center, while a reflection has an entire line (its axis) full of fixed points. In higher math, we find out that one can perform certain reflections and rotations in the coordinate plane using matrix multiplication (which require that the origin be a fixed point), but translations require addition instead.

Also, the fixed points of reflections and rotations allow polygons to have either reflectional or rotational symmetry -- but not translational symmetry. We expect a symmetry of a regular polygon to have at least fixed point -- namely the center of the polygon. As it turns out, only infinite figures (tessellations) can have translational symmetry.

And so I present Section 6-3 of my U of Chicago text, rotations. We start with a definition:

A rotation is the composite of two reflections over intersecting lines.

Some texts define rotations the way you'd expect them to be defined -- in terms of a center and angle of rotation -- and then prove that the composition of two reflections is a rotation. But here, we define rotations to be that composite, then prove that it has the properties you expect. That theorem is called the Two Reflection Theorem for Rotations:

If m intersects l, then the composite of the reflection over m following the reflection over l "turns" figures twice the non-obtuse angle between l and m, measured from l to m, about the point of intersection of the lines.

The text provides the following proof. We are reflecting over lines l and m, and C is the point where they intersect. A is an arbitrary point that we're reflecting. We first reflect A over l to obtain A*, then reflect A* over m to obtain A'. D and E are arbitrary points on lines l and m, respectively, and are used to form angles.

Proof:
(1) C is on both reflecting lines. So C' = C* = C.
(2) By the Figure Reflection Theorem, angle ACD reflected over l is angle A*CD and A*CE reflected over m is angle A'CE. Since reflections preserve angle measure, angles ACD and A*CD have the same measure (call it x) and angles A*CE and A'CE have the same measure (call it y). From the Angle Addition Postulate, the measure of angle DCE is x + y and the measure of angle ACA' is 2(x + y). By substitution, angle ACA' has twice the measure of angle DCE.
(3) Since AC = A*C and A*C = A'C (reflections preserve distance), AC = A'C. QED

In my proof, I prefer to use A' and A" to represent the two images, to emphasize that we are performing two reflections. As in many texts, positive is counterclockwise and negative is clockwise, so we can simply used signed numbers to represent the angle of rotation.

Notice that in preparation for Wu, we especially want to emphasize 180-degree rotations. By the Two Reflection Theorem for Rotations, to obtain a 180-degree rotation, the angle between the two reflecting lines must be half of 180, or 90 degrees. This is why many of my examples have perpendicular reflecting lines.

For these exercises, I include many exercises from Lesson 6-3, but I throw out most of the review Questions, since these are review from mostly Chapter 5 and the first part of Chapter 6, which we obviously haven't covered yet. I only include Question 23 -- but it's a tricky question about the hands of a clock (yet it's relevant here as the clock's hands are actually rotating). I also include the bonus (Exploration) Question 24. Students will have to figure out that if the rotation image of a point A is the point B, then any point equidistant from A and B -- that is, any point on the perpendicular bisector of AB -- can serve as the center of the rotation.