Is $\overline{AB} \cong \overline{BA}$ usually taught as an instance of the symmetric property of congruence?

Question

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?

Answer 1

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.)

Answer 2

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."