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I have been tutoring a wide range of math subjects for many years. Recently, I began tutoring a girl in high school geometry (in California, for context). This semester of the course is starting with a pretty standard introduction to geometric proof writing. Her class has learned several relevant concepts including the three properties of equivalence relations (reflexive, symmetric, and transitive), and they are now beginning the section on the triangle congruence theorems (Side-Side-Side, Side-Angle-Side, etc.).

During proof-writing exercises, the teacher has been telling students that statements such as $\overline{AB} \cong \overline{BA}$ are instances of the symmetric property of congruence. I was taught that congruence was an equivalence relation between geometric objects, and therefore, the way that a geometric object is labelled is irrelevant (at least for geometric objects that have no directionality - like line segments). Additionally, I consider the symmetric property to be a conditional statement that requires a premise (e.g. $\overline{AB} \cong \overline{CD}$) to then draw a conclusion (viz. $\overline{CD} \cong \overline{AB}$). As a result, I would consider the statement $\overline{AB} \cong \overline{BA}$ to be an instance of the reflexive property because (a) it relates a geometric object to itself and (b) it is a simple statement rather than a conditional.

I approached the teacher about this since I was certain it must have been a misunderstanding by my student, but the teacher insisted that only statements like $\overline{AB} \cong \overline{AB}$ should be considered reflexive. I understand where this idea might come from since statements like $\triangle{ABC} \cong \triangle{BCA}$ are not true in general. However, such triangle congruences have an implied ordering or directionality that does not exist for a line segment.

Obviously, I will tutor my student to follow her teacher's instructions (even though I disagree with them), but it makes me curious: Is this the standard way in which the symmetric property of congruence is taught in the US? If not, would this be an example of the reflexive property or some other property?

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  • $\begingroup$ This seems like a detail of the foundational setup. Doesn't the student have a textbook that specifies this kind of thing? Your statement about labeling is reasonable, but I can imagine students being confused because ABCD is not congruent to ACBD. $\endgroup$ – Ben Crowell Feb 9 at 19:40
  • $\begingroup$ @BenCrowell The course does not have a textbook. In fact, because of the ongoing school closures in California, I have gathered from her parents that she barely has a teacher. Online learning has been a very challenging transition. $\endgroup$ – Geoffrey Feb 9 at 20:36
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For me, this is more of a foundational issue than one of secondary education. I would suggest that $\overline {AB}=\overline {BA}$ is a matter of equality rather than congruence, given the definition of line segment as the locus of points between $A$ and $B$. (Strictly speaking, I suppose, it is the locus of points $C$ such that $AC+BC=AB$.) Based on this, it would follow that $\overline {AB}\cong\overline {BA}$ follows from the reflexive principle since they are the same line segment.

I can appreciate the teacher's point to a degree. For instance, if you are showing that $\triangle ABC\cong\triangle DCB$, you could make an argument that the reflexive property statement should be $\overline{BC}\cong\overline{CB}$ even if you were going to go on to justify it with the reflexive property. But I've never known anyone before who used or demanded that level of pedantry outside of maybe computer-based proof checkers.

That said, justifying that statement by the symmetric property of congruence is not valid. The symmetric property of congruence states that if $X\cong Y$, then you can infer $Y\cong X$. The statement that two line segments are symmetric to one another is so nonsensical that it isn't even wrong.

(Disclaimer: I studied and teach in New York state, so I don't know about California. As I have mentioned before, the CCSS don't have much to say about the details of exposing students to proofs, so California could arguably have struck out on its own here.)

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  • $\begingroup$ For what it's worth, that sort of technicality with the triangles is exactly what the teacher does insist on. The expected justification in these proofs is sometimes reflexive and sometimes symmetric depending on exactly what the congruence statement looks like. Based on your answer, I take it that this is not normal in your experience. Additionally, this student is actually at a charter school in California, and I happen to know that this school follows the curriculum posted on the EngageNY website (although they also lean heavily on other sources for specific assignments). $\endgroup$ – Geoffrey Feb 9 at 16:43
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    $\begingroup$ I think calling this pedantry gives too much credit to the teacher, since it implies that the teacher actually understands the issue and is just idiosyncratic (I also think it's just plain wrong instead of pedantic, but that's another issue). This sounds like a typical case of a teacher with weak content knowledge. $\endgroup$ – Thierry Feb 9 at 17:02
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    $\begingroup$ @Geoffrey I haven't taught geometry from EngageNY, but I would be greatly surprised if they were the source of either of these notions. I have never seen a NY Regents exam sample proof docked or commented on for anything we're talking about here. $\endgroup$ – Matthew Daly Feb 9 at 17:07
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    $\begingroup$ @Thierry To be clear, I think that requiring the statement $\overline{AB}\cong\overline{BA}$ when $A$ and $B$ map to each other by the rigid transformation under consideration is what is pedantic. I agree that the misunderstanding of the symmetric property of congruence (which is actually that $X\cong Y$ implies $Y\cong X$) is weak content knowledge. $\endgroup$ – Matthew Daly Feb 9 at 17:12
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    $\begingroup$ @Thierry Actually, I edited my response based on your comment to be more clear. Well observed! $\endgroup$ – Matthew Daly Feb 9 at 17:50
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In axiomatizations of Euclidean plane geometry such as the ones by Hilbert or Tarski, the statement $\overline {AB}\cong\overline {BA}$ is a postulate. In Tarski's system, this congruence axiom is explicitly called "Reflexivity of Congruence."

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