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.
$\endgroup$ 33 Answers
$\begingroup$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.
$\endgroup$ 3 $\begingroup$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.
$\endgroup$ 4 $\begingroup$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.
$\endgroup$ 1
