So the problem I have is to write all the properties that a relation has (reflexive, symmetric, transitive, irreflexive, antisymmetric). The problem is the congruence relation on the set of triangles.
I know it is reflexive because any triangle is of course congruent with itself.
For symmetric I am confused. If you have triangle $ABC$ that is congruent with triangle $DEF$, then $DEF$ is congruent with $ABC$. But is this true if they are not equal? Like, to prove antisymmetry you must show that $xRy$ and $yRx$ to where $x = y$. $ABC$ isCongruentTo $DEF$ so then $DEF$ isCongruentTo $ABC$, but $ABC = DEF$ so therefore antisymmetric? Can something be both symmetric and antisymmetric?
The other properties I feel confident with.