Updated 2013 June 15.

This is my appeal to my colleagues, and to the mathematical community, to stop using the acronym “CPCTC.” Here’s my case against CPCTC:

- Students cannot consistently recite the sentence that “CPCTC” abbreviates.
- The sentence itself sounds awkward and wordy: “The corresponding parts of congruent triangles are congruent.” Since the last three words are “triangles are congruent,” many students erroneously believe that the statement says that triangles are congruent. This is much the same error as saying that “the pockets of my jeans are dirty” is about dirty jeans, as opposed to dirty pockets.
- Students cite CPCTC at the wrong times. The classic error here is to assert that two triangles are congruent “by CPCTC.” This shows a lack of understanding on the part of the student.
- CPCTC tries to do two things at once: by means of the term “parts,” it tries to subsume statements about *angles* as well as *sides.”

The solution I propose is simple: instead of trying to do double-duty by using the term “parts,” let’s separate this idea into the following two statements:

- CT->CS: Congruent triangles have congruent sides. Or, if you prefer: “If two triangles are congruent, then they have congruent sides.”
- CT->CA: Congruent triangles have congruent angles.

Do you notice the same problems I mention above? What do you think of my proposed solution? Let me know what you think about this idea in the comments.

### Like this:

Like Loading...

*Related*

I agree with absolutely everything you’ve written here. Insisting that students use the phrase “CPCTC” is just teacher-on-student posturing: “Can you apply this awkward and wordy expression that makes us both sound very intelligent? Yes? Good, now, let’s try it in Latin!”

More fun facts about CPCTC: only 5 important words (and two short linking words), but a healthy 16 syllables!

I have also observed students using CPCTC incorrectly, applying it in completely innappropriate situations and revealing that they have no idea what the phrase means. That’s because the acronym is designed so that you don’t have to think about what it means. Would you insist that your students write the entire phrase “corresponsing parts of congruent triangles are congruent” on their paper? Of course not, it’s a long-winded way of expressing a very simple concept. So the acronym is proposed as a way of expressing the idea without having to write all those words. But we are not just lazy with our writing, we are lazy with our thinking, as well! Do the students think of the phrase as they write the acronym? No! They write CPCTC (which rolls right off the tongue) and think nothing at all. If your expression is so annoying that students (and teachers) are too lazy to even think it, then maybe you should look around for a better expression.

Of course, if you are writing things down and thinking nothing at all, then the entire activity of constructing a meaningful proof has been subverted.

Another comment… We are not consistent in our application of these kinds of locutions. When using a triangle similarity statement to justify a proportion, do we write CSSTP (corresponding sides of similar triangles are proportional) or, for the angles, CASTC (corresponding angles of similar triangles are congruent)? Of course we don’t. We just say “because the triangles are similar” or, with the textbook we use, “polygon similarity postulate”.

Another comment… What’s most important in an application of CPCTC, in my opinion, is that the students connect their congruence statement to a previously established congruence between triangles. And so, in an actual proof, it would be better for them to write, next to a congruence statement, as a justification, something like “Tri ABC =~ Tri DEF”. Or, if you like numbering your lines, the justification could just be “line 4”, which refers to the line where the appropriate triangle congruence has been established.

Citing the line numbers forces students to confirm that they actually have the necessary information to make the inference that they are trying to make. This is ostensibly the reason that we make students give justifications for their statements — because it helps them think through the proof and convinces them that their reasoning is correct. (Apart from other other reason to make them do it — because it helps the teacher grade the proof). So, in conclusion, make your students cite line numbers in proofs, if you make them give reasons at all.

Last comment… The larger issue here is that mathematics (and geometry in particular) has a lot of technical language that students are forced to master. In addition to the useful vocabulary that is standard in the mathematical community (point, line, plane, angle, ray, segment, rhombus, parallelogram, locus, bisector, tangent, inscribed angle, etc.), students are forced to master a great deal of other vocabulary that is not standard, such as linear pair postulate, side-splitting theorem, and polygon similarity postulate. These differ widely from textbook to textbook and are chosen for their simplicity. It’s not really necessary for students to know the names of these theorems, but we feel like there should be a name, so we impose a standard for the sake of consistency. But there are way too many theorems for every one to get a catchy name! For example, the theorem that says that if a quadrilateral has a pair of opposite sides that are both congruent and parallel, then it is a parallelogram. What are we to call this? Some textbooks just call it Proposition 4.4.1 or something like that. I tell my students to just write “property of parallelograms” or some such abbreviation (“prop of p-gram”), other teachers will have their students write out the entire sentence.

One thing that I’ve tried in the past is to let the students pick their own names for things that are not standard. When a new thing (like CPCTC) is first introduced, let them come up with their own name for the justification: “Duh, triangles are congruent” (DTAC) or “Triangles are congruent, dog.” (TACD) or something else ridiculous like that. I find that it is worth the 3-4 minutes of class discussion and temporary chaos to agree to a phrase that the students feel they have some input into. Then you just have a class vote and that becomes the standard for the rest of the semester. It’s just human nature to be more engaged in something if you get to name it yourself. And teenagers are at a point in their lives of maximum linguistic innovation. They are literally reinventing the English language each generation! The phrase “CPCTC” is not only long and annoying, it’s old fashioned, and therefore lame. But if the students are allowed to write on their paper MC-TriP (“My Congruent TRIangles (have been) Proved”), that’s way, way more fun and forces them to think about what they are saying.

I am going to be totally bold here (as someone who went through a proofs based high school class, and then went on to get a math degree, and never heard the term CPCTC until recently) that we can extend saying “by congruence” as a good enough explanation where everyone will understand what it means.

Jason, thanks for your comment. I think the root cause of the problem in this area is the push to abbreviate everything, presumably for the sake of efficiency in writing 2-column proofs. Solution: stop abbreviating, and stop the practice of writing 2-column proofs! Why don’t we let students express their own complete thoughts in their own complete sentences? “…Therefore ABC and DEF are congruent by the SSS Rule, which means that side AB must be congruent to side DE.” That would be good enough for me, and I think abundantly clearer than writing “CPCTC!”