Tuesday, November 18, 2014

Section 6-7: Corresponding Parts in Congruent Figures (Day 66)

Tonight I will tutor my geometry student.

Section 6-7 of the U of Chicago text covers the Corresponding Parts in Congruent Figures Theorem, which the text abbreviates as CPCF. But a special case of this theorem is more widely known -- corresponding parts in congruent triangles are congruent, or CPCTC.

When I was young, a local PBS station aired a show called Homework Hotline. After school, middle and high school students would call in their homework questions in math and English, and some would be chosen to have their questions answered on the air by special teachers. Even when I was in elementary school, I often followed the geometry proofs that were called in, and more often than not, there were triangle congruence two-column proofs where the Reason for a step was often CPCTC. So this was where I saw the abbreviation CPCTC for the first time. (By the time I reached high school, a few calculus problems were called in to the show. Nowadays, with the advent of the Internet, the show has become obsolete.)

Here's a link to an old LA Times article about Homework Hotline:


When I reached geometry, our text usually either wrote out "corresponding parts in congruent triangles are congruent," or abbreviated as "corr. parts of cong. tri. are cong.," probably with a symbol for congruent and possibly for triangle as well. But our teacher used the abbreviation CPCTC. Now most texts use the abbreviation CPCTC -- except the U of Chicago, that is. It's the only text where I see the abbreviation CPCF instead.

Dr. Franklin Mason, meanwhile, has changed his online text several times. In his latest version, Dr. M uses the abbreviation CPCTE, "corresponding parts of congruent triangles are equal."

Well, I'm going to use CPCTC in my worksheets, despite their being based on a text that uses the abbreviation CPCF instead, because CPCTC is so well known.

Once again, it all goes back to what is most easily understood by the students. Using CPCTC would confuse students if they often had to prove congruence of figures other than triangles. But as we all know, in practice the vast majority of figures to be proved congruent are triangles. In this case, using CPCF is far more confusing. Why should students had to learn the abbreviation CPCF -- especially if they have already seen CPCTC before (possibly by transferring from another class that uses a text with CPCTC, or possibly even in the eighth grade math course) -- for the sole purpose of proving the congruence of non-triangles, which they'd rarely do anyway?

So it's settled. On my worksheet, I only use CPCTC.

Notice that for many texts, CPCTC is a definition -- it's the meaning half of the old definition of congruent polygons (those having all segments and angles congruent). But for us, it's truly a theorem, as it follows from the fact that isometries preserve distance and angle measure.

In this lesson, students basically learn what CPCTC is. Of course, they won't actually use it in any proofs until Chapter 7. Notice that my lesson begins with the same example as the text, where we begin with a triangle and then translate it, reflect it, and then rotate it. I begin the same way, except I do it in a different order -- translation, then rotation, and then reflection. This is because when given two congruent triangles and one wants to map one to the other, the most intuitive way to do so is to translate it so that one pair of vertices coincides, then rotate it so that one whole side coincides, and if that's not enough to make the entire triangles coincide, then one final reflection will do the trick. This is, in fact, how Dr. M proves SAS. (This is also why my triangle images overlap -- I want to think in terms of mapping a preimage onto a target image, one isometry at a time.)

This also leads to another question. We've made a big deal about how some results require the Fifth Postulate and how others don't. Technically speaking, SSS, SAS, and ASA are all true in all three types of geometry -- Euclidean, hyperbolic, and spherical. (AAS and HL, on the other hand, do require the Fifth Postulate. This is why Dr. M introduces SSS, SAS, and ASA in his Chapter 3, but he had to wait until Chapter 4 to give AAS or HL) So we could have taught SSS, SAS, and ASA much earlier, before we covered the Fifth Postulate. But not only does this modify the U of Chicago order, but it means that we can't use translations, whose properties derive from the Fifth Postulate. Of course, we could just use reflections instead, but translations are much easier to understand. When we want to map triangle ABC onto a congruent triangle DEF, it's much easier to begin by translating A to D, rather than reflecting over the perpendicular bisector of AD. This is why we're waiting so late before introducing SSS, SAS, and ASA.

Another issue that comes up is the definition of the word "corresponding." Notice that by using isometries, it's now plain what "corresponding" parts are. Corresponding parts are the preimage and image of some isometry. Unfortunately, we use the word "corresponding angles" to mean two different things in geometry. When two lines are cut by a transversal and, "corresponding angles" are congruent, the lines are parallel, but when two triangles are congruent, "corresponding angles" (and sides) are congruent as well. The phrase "corresponding angles" has two different meanings here! Of course, one could unify the two definitions by noting that the corresponding angles at a transversal are the preimage and image under some isometry. I tried this earlier, remember? It turns out that the necessary isometry is a translation, whose properties depend on the Fifth Postulate, yet we want the Corresponding Angles Test (not the Consequence, but the Test) to be proved without using any Fifth Postulate at all.

So unfortunately, we're stuck with two unrelated definitions of "corresponding angles."

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