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Why use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) instead of just "Definition of Congruent Figures" especially since definitions are biconditional?

I'm working on high-school level Geometry and specifically "reasons" in two-column statement-reason proofs.

tuna
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    What is your definition of "congruent figure"? – rschwieb Jul 08 '15 at 17:03
  • From the book (Geometry, McDougal Littell, 2007, pg. 225): "In two congruent figures, all the parts of one figure are congruent to the corresponding parts of the other figure. In congruent polygons, this means that the corresponding sides and the corresponding angles are congruent." – tuna Jul 14 '15 at 15:16
  • If this is true, then yes, CPCTC would just be an a priori conclusion from the definition of congruent triangles. But I expect that this is some miscommunication here, and that "congruent triangles" are defined in some more interesting way, and then the statement of CPCTC that you describe gains some nontriviality. – rschwieb Jul 14 '15 at 15:46

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Note that this answer is a speculative / philosophical one at best.

I suppose that the point of such statement-reason proofs is to get a high school mind to think more precisely about mathematics. I think here the teacher / curriculum wants to get the student to think about what specific salient property is involved in the assertion they make about congruent sides or angles.

In this way, you are also refreshed about what makes two figures congruent - namely, that each of their corresponding parts (sides and angles) are congruent to each other.

Had you only needed to write "Definition of congruent figures," it could be argued that that would have perpetuated the same formulaic type of math that so many of us dislike about the current state of young mathematics education. The intuitive meaning of "congruence" might be lost in the mind of a student who just memorizes what phrase to put for what reason in the proof.

MT_
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  • BTW, the word you want is "speculative", not "speculatory". – Robert Israel Jul 08 '15 at 16:59
  • @RobertIsrael Thanks, haha. – MT_ Jul 08 '15 at 17:01
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    Okay, so no "strict" mathematical reason to use one over the other, but one may promote mathematical thinking while another emphasizes mathematical elegance, and both may encourage a student to think more deeply about the topic at hand. Like you said, a more philosophical question; thanks for your answer anyway to clarify that. – tuna Jul 14 '15 at 15:23
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Another speculatory one.

I think the emphasis might be on the "Corresponding" part of the name, since it's a common mistake to equate not corresponding parts of the triangles in hand. For that reason, having it in your reasons column presumably makes you check it?

Then, of course, the abbreviation clearly negates that effect, but it can still be the reason the textbooks use it.

Dosidis
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Lots of theorems in mathematics assert the equivalence of two collections of conditions. That is apparently the function of the "CPCTC" theorem you are referring to here.

By learning two different versions of the same concept, you gain some insight into it. In some circumstances, one of the collections of axioms might be easier to use compared to the other set. It is not surprising to have even several biconditionally equivalent collections of hypotheses. Each re-expression of the concept offers a different perspective.

Several advanced examples come to mind immediately, but I hesitate to use them. Let me know if you're still interested, and in the meantime I'll try to think of a more elementary one.

rschwieb
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  • I think I understand what you're saying, at least in a general sense. Thanks for the answer; I may not understand more advanced examples, but post them if you think they'll "finish" the discussion here! – tuna Jul 14 '15 at 15:26