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My Geometry class is doing triangle congruency proofs these days. In general, we find three pairs of congruent parts (sides or angles) in two triangles; we show that these congruencies reveal that the triangles are either congruent or similar; and we conclude that further parts are congruent (in the case of congruent triangles) or proportional (in the case of similar triangles).

In the column of justifications, therefore, the last two justifications are usually SSS (or SAS or ASA or AAS or AAA) and then either CPCTC (Corresponding Parts of Congruent Triangles are Congruent) or Definition of Similarity.

I’ve done pretty much the same thing each time I’ve taught Geometry, but this year we’re using a text by Pearson in which this topic appears in Chapter 4. After doing this a zillion times, I have begun to wonder something about this nomenclature.

Why is that where triangles are CONGRUENT, we DO refer to CPCTC, and we DO NOT refer to a definition; but where triangles are SIMILAR, we DO NOT refer to CSSTP, but we DO refer to a definition?

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  • $\begingroup$ Suggest reordering the last paragraph (and anything related) to be in the same order ("first instance"/"second instance") as the attached chart. Currently it's a bit confusing which of the two orders is being referenced. $\endgroup$ – Daniel R. Collins Nov 30 '17 at 16:04
  • $\begingroup$ I think that's helpful, thanks. $\endgroup$ – Daniel R. Collins Dec 1 '17 at 4:06
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Put me in the camp of people who would prefer to say "Definition of Congruent Triangles" rather than "CPCTC". However, even if we agree that CPCTC is too well-established in the vernacular to get rid of, there are still good reasons not to use CSSTP: it is too specific and does not cover all of the different uses for which triangle similarity proofs are used. Consider the following triangle similarity arguments:

  • Use two pairs of congruent angles to prove triangles similar; then conclude that the sides are pairwise-proportional. ("Corresponding sides of similar triangles are proportional", or CSSTP.)
  • Use two pairs of pairwise-proportional sides and one pair of congruent included angles to prove triangles similar; then conclude that the third pair of sides is in the same ratio as the others. ("Corresponding sides of similar triangles are proportional", or CSSTP.)
  • Use two pairs of pairwise-proportional sides and one pair of congruent included angles to prove triangles similar; then conclude that the other two pairs of angles are pairwise-congruent. ("Corresponding angles of similar triangles are congruent", or CASTC.)
  • Use three pairs of pairwise-proportional sides to prove triangles similar; then conclude that all three pairs of angles are pairwise-congruent. ("Corresponding angles of similar triangles are congruent", or CASTC.)

"Definition of Similarity" encompasses both CSSTP and CASTC. The distinction is not necessary to make when discussing triangle congruence, because "congruent angles" and "congruent sides" are described using the same word (even though they are technically different relations).

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  • $\begingroup$ Those are great points. I had clearly not thought of them, and I had to think them over for several minutes. Your point is that one term is well-chosen and one is not, and we cannot change that now. Right? When CPCTC was introduced, of course, it was not yet well-established. What could have led to that choice then? $\endgroup$ – Chaim Dec 1 '17 at 14:16
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In a mathematical proof, any logically valid justification is permissible. In particular, you can cite any definition, assumption, or previously proved theorem at any time. Also, strictly speaking, you never have to appeal to a theorem, since a theorem is just a consequence of the definitions; the point of packaging reasoning in theorems is to avoid duplication of effort, organize concepts, and make proofs shorter and easier to read.

So, to directly answer your question, for both similarity and congruence, you can either prove something straight from the definition or cite a relevant theorem. If your textbook seems to prefer one over the other, that's probably just an arbitrary choice by the author(s), not anything logically necessary. If your textbook makes it seem like only one way is acceptable, that's a serious flaw in the textbook, since it'd be reinforcing the harmful misconception that there's only one way to prove a given theorem.

In the specific context of congruence, by the way, I don't know why you wouldn't just teach the following more general theorem: Rigid motions of the plane preserve measures of angles and lengths of line segments. The proof is exactly the same—assuming you've defined "congruent shapes" in the usual way, that there's a rigid motion of the plane taking one to the other—but the statement doesn't unnecessarily require the line segments and angles to occur as part of triangles. Likewise for similarity: Uniform scaling of the plane preserves measures of angles and ratios of lengths of line segments. I can understand restricting to special cases when the general case is more complicated, but here, there's no extra complexity in general, and I'd be concerned that students would think it's something special about triangles rather than completely general properties of congruence and similarity.

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  • $\begingroup$ Well in my experience teaching in high schools for many years, we always do this the same way: CPCTC and Definition of Similarity, not CSSTS or Definition of Congruence. If the choice is arbitrary, why is it made the same way in many books? Comparable justifications (both by definition or both by CPCTC/CSSTS) would seem more natural to me, rather than this false distinction. $\endgroup$ – Chaim Nov 30 '17 at 18:19
  • $\begingroup$ I think I follow the second point, about generality; but I think that each individual (like the human race in general) usually proceeds from specific examples to greater generality because that's somehow more natural to our psychology. For example Thales "measured the height of the pyramids by their shadows at the moment when his own shadow was equal to his height" (according to Wikipedia) although that moment was not really special either. $\endgroup$ – Chaim Nov 30 '17 at 18:21
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    $\begingroup$ K-12 math textbooks from major publishers in the US are notorious for inventing strange terminology and abbreviations that are basically never used anywhere else (like "CPCTC" and "CSSTS"), making arbitrary notational conventions and quirks of formatting seem like important facts, giving vague definitions or no definitions at all, and picking up on each other's weird terminology and conventions so the same oddities start showing up in a bunch of books (like "two-column proofs", which pretty much only exist in high school geometry class). $\endgroup$ – Daniel Hast Nov 30 '17 at 18:40
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    $\begingroup$ One mathematician and math education researcher, Hung-Hsi Wu, has some good writings about this. He describes school textbooks as having essentially created their own language, which he calls "Textbook School Mathematics", that diverges from the rest of mathematics, doesn't follow certain fundamental principles of mathematical practice, and is in many cases barely recognizable to mathematicians. For example, here's a post where he talks about this: blogs.ams.org/matheducation/2015/02/20/… $\endgroup$ – Daniel Hast Nov 30 '17 at 18:43
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    $\begingroup$ @DanielHast: Some good points, and I love reading Wu; however, I disagree about the critique of a two-column proof. Literally ever math book I've ever read at some point shows some symbolic manipulations with explanatory descriptions noted in the right margin. While the high school geometry bit may be a bit more formalized, IME the basic idea is a nigh-universal practice, and I see nothing shameful in it. $\endgroup$ – Daniel R. Collins Dec 1 '17 at 4:10

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