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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."


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, ...

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