Lesson 4-2: Reflecting Figures (Day 24)

This is what I wrote last year about today's lesson. Last week I mentioned that I was giving a quiz because it was the end of the "quaver" -- in this post, I explain what a "quaver" is. I had to make one major change to this post -- Dr. Franklin Mason has changed his Lesson 3.1 since I first posted this.

Lesson 4-2 of the U of Chicago text moves from reflecting individual points to reflecting entire figures. The cornerstone is the Reflection Postulate. Nearly every theorem involving transformations and congruence can be ultimately traced back to the Reflection Postulate.

If an academic year is 180 days, then a quarter is 45 days. Then today marks the midpoint of the first quarter according to my calendar. Half of the academic year is called a semester, and half of a semester is called a quarter, but what is half of a quarter called?

Twenty-one years ago was the year that I first learned high school geometry, but it was another class -- history -- where I wondered about half of an academic quarter. This was because my school divided the school year into quarters, and midway through the quarter, unsatisfactory notices were due. And therefore the history teacher gave a major test halfway through the quarter, and again at the end of the quarter. And so I wanted to give a name to the period that was half of a quarter.

The name I came up with is quaver. This term comes from music -- half of a quarter note is known as an eighth note, but the British call eighth notes "quavers." And so today marks the end of the first quaver and the start of the second quaver of my academic year. (Of course I'm referring to my Middle Start district -- today I subbed in a science class in my Early Start district, where today is Day 34.)

Of course, this is all illogical nonsense. In music, a quaver isn't half of a quarter -- it's half of a quarter note, and only when we mix American and British terms. In the UK, a quarter note is called a "crotchet," and so if I really want to be consistent, I should call an academic quarter a "crotchet." But I won't. On this blog, I'm going to call half of a quarter, a quaver. The word quaver sounds a lot like quarter, and I like referring to each quaver of the academic year the same way I refer to each quarter, so there!

Of course, we as geometry teachers aren't always linguistically consistent either. Let's look at the names of various terms for polygons:

n-sided polygon (poly-, Greek "many")
5-sided pentagon (penta-, Greek "five")
6-sided hexagon (hexa-, Greek "six")
7-sided heptagon (hepta-, Greek "seven")
8-sided octagon (octa-, Greek "eight")

That's all fine, except when we reach:

9-sided nonagon (nona-, Latin "nine each")

And so we mix up Greek prefixes with Latin prefixes, just as I mix up the American quarter (note) with the British quaver. The logical term for a 9-sided polygon using a Greek prefix is "enneagon." (Notice that back in Lesson 2-7, in the Bonus question, one of the obsolete terms for a 9-sided polygon was "enneagon.")

Incidentally, I do use musical terms for periods longer than a quaver -- though not "crotchet." In Britain, a "breve" is twice as long as an American whole note, while a "longa" is twice as long as a breve. And so I occasionally refer to periods of two- and four-years as breves and longae, respectively. Thus the nominal length of each of the "paths" that I mentioned a few weeks back is a breve, while the nominal lengths of high school and (undergraduate) college are each a longa. (The official names of these two periods are "biennium" and "quadrennium" respectively -- but I like my names better.)

Now returning to the subject of this blog, 21 years ago -- or is that five longae and one year -- ago, I first sat down in a geometry class. But due to the Common Core, things have changed. Many statements that were postulates 21 years ago are now theorems -- most notably SSS, SAS, and ASA. They are provable statements, and the proofs begin with the Reflection Postulate.

Let me state the Reflection Postulate, because it is so important, directly from the U of Chicago text:

Reflection Postulate:
Under a reflection:
a. There is a 1-1 correspondence between points and their images.
This means that each preimage has exactly one image, and each image comes from exactly one preimage.

b. If three points are collinear, then their images are collinear.
Reflections preserve collinearity. The image of a line is a line.

c. If B is between A and C, then the image of B is between the images of A and C.
Reflections preserve betweenness. The image of a line segment is a line segment.

d. The distance between two preimages equals the distance between their images.
Reflections preserve distance.

e. The image of an angle is an angle of the same measure.
Reflections preserve angle measure.

This postulate corresponds to Dr. Franklin Mason's "Rigid Motion Postulate," in the old version of his Lesson 3.1 last year. Since then, Dr. M has completely changed his Chapter 3 -- this is almost certainly because isometries ("rigid motions") aren't emphasized on the Common Core texts nearly as much as either of us thought they would when we first read the standards. Nowadays, Dr. M uses the classical definition of congruent polygons (i.e., equality of corresponding measures). He assumes SAS as a postulate (just as the mathematician Hilbert did a century ago), and uses Euclid's ancient proof to derive ASA. But for SSS, Dr. M still uses rigid motions to move one of the triangles into place (similar to the start of the U of Chicago proof) before using SAS and Isosceles Triangle Theorem to prove SSS.

Part a is a very important part of the Reflection Postulate. Without it, a point A could have two reflection images -- there could be two distinct points B and C such that the reflecting line m is the perpendicular bisector of both AB and AC. (I believe that the Ruler and Protractor Postulates, or their U of Chicago equivalents, are sufficient to prove the existence of at least one reflection image, but to prove that at most one reflection image exists requires the Reflection Postulate.) This was a problem for Dr. Hung-Hsi Wu, who decided to define rotation before reflection and then use the properties of rotations to prove that every point has at most one reflection image. But since I want to use reflections to define rotation, I am forced to assume that reflection images uniquely exist as part of the postulate to avoid circularity.

According to the text, reflections preserve:
Angle measure
Betweenness
Collinearity
Distance

a nice little mnemonic for the students.

The first theorem of this chapter is the Figure Reflection Theorem:
If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points.

This theorem is used to conclude, for example, that if A' is the image of A, and B' is the image of B, then A'B' is the image of AB. An informal proof is given. A formal proof is a little bit tricky -- sure, we know that if C is between A and B, then C' is between A' and B'. But the problem is the converse -- if there's a point D between A' and B', how do we know there's a point C between A and B such that C' is exactly D? The best way is probably to use the Distance part of the Reflection Postulate -- we choose C to be the point on AB such that AC = A'D (which exists by Ruler Postulate). Then since reflections preserve distance, C' must be exactly D.

But that is the sort of proof that I don't want to confuse students with. It's best just to use the informal proof given in the text and save formal proofs for later.

The text states that when a figure intersects the reflecting line, the image must intersect the reflecting line in the same point or points. This follows immediately from the fact that the image of a point on the reflecting line is the point itself.

But what I find interesting is the related statement -- if a figure intersects its reflection image, then it must intersect the reflecting line in the same point or points. This statement is false in general, but it's true if the figure to be reflected is itself a line. This fact helps us greatly -- for example, consider Question 21 from the text:

The reflection image of Triangle ABC is Triangle XYZ. Now Lines AC and XZ intersect at a point -- which we now know must lie on the reflecting line. And Lines BC and YZ intersect at a point -- which we now know must lie on the reflecting line as well. And those two points determine exactly one line -- the reflecting line! So all the student has to do is draw the line through the two points of intersection.

And as a corollary, it follows that if a line is parallel to the reflecting line, it must be parallel to its reflection image, Last year, I called this the "Line Parallel to Mirror Theorem" (where "mirror" refers to the reflecting line) But I will wait a few days before introducing that theorem to students. (Notice that the Common Core Standards state that a line must be parallel to its dilation image, so why not give the conditions when a line is parallel to its reflection, rotation, or translation images?)