Section 12-9: The AA and SAS Similarity Theorems (Day 91)

Section 12-9 of the U of Chicago text covers the AA and SAS Similarity Theorems. The reason that I cover these two first and skip over SSS Similarity is because of Dr. Wu. For he states the theorems in the reverse order from the U of Chicago -- in the text SSS is first, then AA, then SAS, while for Wu, SAS is Theorem 26, AA is Theorem 27, and only afterward does SSS appear.

This section gives a proof of AA and asks the students to prove SAS. Once again, this is in stark contrast to most texts where AA is a postulate. So let's take a look at the proof of AA Similarity as given in the U of Chicago text. I'll keep this one in paragraph form for now, as it is in the text, but as usual, I made cosmetic changes such as replacing "size transformation" with "dilation":

AA Similarity Theorem (Wu's Theorem 27):
If two triangles have two angles of one congruent to two angles of the other, then the triangles are similar.

Given: Triangles ABC and XYZ with Angle A = X and Angle B = Y.
Prove: Triangle ABC ~ XYZ.

Proof:
Since XY and AB are corresponding sides, let k = XY/AB be the magnitude of a dilation applied to Triangle ABC. Then A'B' = k * AB = XY/AB * AB = XY. Also, since dilations preserve angle measure, Angle A = A' and Angle B = B'. With transitivity, Angle A' = X and Angle B' = Y. So Triangle A'B'C' is congruent to XYZ by the ASA Congruence Theorem. Thus Triangle ABC can be mapped onto XYZ by a composite of dilations and reflections. So Triangle ABC ~ XYZ. QED

Notice that in some ways, Wu's definition of similar works better here -- we want to show that there exists some dilation D such that D(ABC) is congruent to Triangle XYZ. The above proof finds exactly such a dilation -- namely one with scale factor XY/AB. As it turns out, the center of the dilation is irrelevant -- any dilation with the correct scale factor will work.

I won't bother posting the Wu proof, as it's nearly identical to this one, except with different notation (in particular, Wu uses A'B'C', not XYZ, to denote the triangle to be proved similar to ABC, and so he comes up with notation like A*B*C* or A0B0C0 to denote the dilation image of ABC).

The U of Chicago text directs the students to prove SAS Similarity in Question 11, and I will likewise include it on my worksheet an an exercise for students to complete. Since 11 is an odd number, this question is included with the answers in the back of the book. Here is the proof:

Given: Triangles ABC and XYZ with Angle A = X, AB/XY, AC/XZ.
Prove: Triangle ABC ~ XYZ.

Proof:
Let k = XY/AB, and then find Triangle A'B'C' ~ ABC with scale factor k. Then
A'B' = k * AB = XY/AB * AB = XY and A'C' = k * AC = XY/AB * AC = XZ/AC * AC = XZ. So Triangle A'B'C' is congruent to XYZ by the SAS Congruence Theorem. Thus Triangle ABC can be mapped onto XYZ by a composite of dilations and reflections, so Triangle ABC ~ XYZ. QED

Notice that neither Wu nor the U of Chicago choose any arbitrary dilation with the correct scale factor -- both choose the dilation centered at exactly the point A with the correct scale factor. This makes the proof easier to understand, as A is actually the same point as A' -- so Angles A and A' are congruent by the Reflexive Property.

As I mentioned before, I've seen pre-Common Core texts that give a proof of the SAS Similarity Theorem, yet declare AA to be a postulate. It appears what these are doing is proving SAS Similarity using AA Similarity in lieu of dilations. First B' is chosen on AB so that A'B' = XY (Ruler Postulate), and then C is chosen so that B'C' | | BC (Playfair), and then the Corresponding Angles Consequence gives enough congruent angles to conclude ABC ~ A'B'C' by AA. Then one can prove A'B'C' and XYZ congruent by SAS Congruence, just as in the Wu and U of Chicago proofs.

The text gives one example of a proof using SAS Similarity, written in two-column format:

Given: T is the midpoint of PS; Q is the midpoint of PR.
Prove: Triangle PTQ ~ PSR.

Proof:
Statements                              Reasons
1. T, Q midpts. PS, PR            1. Given
2. PT = 1/2 PS, PQ = 1/2 PR  2. Definition of midpoint
3. PT/PS = 1/2, PQ/PR = 1/2  3. Multiplication Property of Equality
4. Angle P = P                        4. Reflexive Property of Congruence
5. Triangle PTQ ~ PSR           5. SAS Similarity Theorem (steps 3, 4)

Recall that in the U of Chicago, there's no Division (or Subtraction) Property of Equality, so we should think of deriving PT/PS = 1/2 by multiplying by 1/PS, not dividing by PS.

Here are a few more things that I want to say about similarity. We know that for congruence, there are only four isometries -- reflections, rotations, translations, and glide reflections. Now a similarity transformation is the composite of a dilation and an isometry. So there are at least five similarity transformations, but these clearly aren't all of them, since the composition of a dilation and certain types of isometries is neither an isometry nor a dilation.

As it turns out, the composite of a dilation and a translation is another dilation, only with a different center but the same scale factor as the original dilation. The same thing happens if we replace "dilations" with "rotations" above -- the composite of a rotation and a translation is another rotation, with a different center yet same angle. But that rule (rotation and translation equals rotation) only works in two dimensions, but the former rule (dilation and translation equals dilation) works not just in 2D, but in all dimensions.

In fact, believe it or not, some (college-level) texts actually define dilation so that translations are included as dilations! These use the word homothety to denote the strict dilations -- that is, those that aren't translations. In a hierarchy, similarity transformations would have dilations as one of its branches, and dilations in turn would have two branches, homotheties and translations. Of course, the identity would still be both a homothety and a translation.

I don't use the word homothety in any of my lessons, nor do I consider translations to be dilations, since both of these would confuse high school students.

This means that there are two transformations -- reflections and rotations -- whose composite with a dilation is not another dilation. If one wanted to, we could give the composite of a rotation and a dilation a special name, a roto-dilation. I guess this would make the composite of a reflection and a dilation something strange sounding like reflecto-dilation. Now what would happen if we were to compose a glide reflection with a dilatio?. Notice that since a glide reflection is the composite of a reflection and a translation, the dilation absorbs the translation, leaving the composite of a reflection and a dilation. Thus, all orientation-reversing similarity transformations that aren't isometries must in fact be "reflecto-dilations."

In all, there are seven similarity transformations: reflections, rotations, translations, glide reflections, dilations, roto-dilations, and reflecto-dilations.

All of this, of course, presumes Euclidean geometry. In non-Euclidean geometry, there are, in fact, no dilations, and so all similarity transformations are in fact isometries! Furthermore, there are some forms of non-Euclidean geometry that contain transformations that look like dilations, but are, in actuality, translations!

Yesterday, I mentioned Danica McKellar's Girls Get Curves. Well, there was another book that I purchased over the weekend: Simon Singh's The Simpsons and Their Mathematical Secrets.

I've mentioned some of the math that's appeared on the TV show The Simpsons before. When Lisa helps her brother Bart use geometry to win a miniature golf tournament, this appears in Chapter 6, "Lisa Simpson, Queen of Stats and Bats." Actually math has appeared several times on the show throughout the quarter-century that it's been on TV -- this is because the shows writers include several with advanced Ivy League degrees in math. The very same day that I bought the book, the episode that night made a brief reference to the college-level math theorem, Zorn's Lemma. (This theorem is notable for requiring the Axiom of Choice -- set theory's Parallel Postulate -- in its proof.)

Like McKellar, Singh also comments on girls' interest in math. In the Season 17 Episode "Girls Just Want to Have Sums," Lisa wants to prove that she's the best math student in the school, but her male principal, Mr. Skinner, discourages her. The school ends up separating into a boys' school and a girls' school, and the girls' math class ends up teaching touchy-feely stuff -- Lisa ends up dressing as a boy and sneaking into the male school. Singh writes, in Chapter 7:

"The division between feminine and masculine mathematics is only fictional, but it echoes a real trend in recent decades towards touchy-feely mathematics for both boys and girls. Many members of the older generation are concerned that today's students are not being stretched in terms of tackling traditional problems, but instead are being spoon-fed a more trivial curriculum."

Of course, this sounds like Common Core. But notice that Singh wrote "recent decades," so this was a problem long before Common Core. I won't write Singh's example of how a math problem has changed since the 1950's, but here's a link to the list it's based on:

http://www.crazystudent.com/practical-jokes/teacher/evolution-of-math-problem.htm

I've already stated why one might want to water down math over the decades -- to protect those who know all the math that one needs to know in real life, which is far short of Algebra I, from being denied a high school diploma. Once again, there must be some middle ground between forcing all students to learn calculus in senior year and giving students like Lisa "feminine math."

(An aside: In Chapter 5, "Six Degrees of Separation," Singh writes that he's well-connected to both the Hungarian mathematician Paul Erdos and the American actor Kevin Bacon, such that Singh has an Erdos-Bacon number of six: four for Erdos, two for Bacon. As it turns out, Danica McKellar also has an Erdos-Bacon number of six, divided the same was as Singh.)

For me, the take-home message from both McKellar and Singh is, make sure that I treat my male and female students equally in my math classes, and not become another Principal Skinner.