There was a movement in the 1800s to replace the Elements with a more modern textbook and a number of authors used different definitions.
What other definitions of “parallel line” do you have in mind?
Is it obvious from the definition of parallel l lines that this ought to be true? That equality should be transitive seems like so obvious an idea that it’s barely worth writing down.
Congruence and equality are not the same thing. One of these axioms says that being parallel is transitive; the other says that being congruent is transitive. I agree that both notions become much less useful if transitivity does not hold, but a non-transitive congruence relation is not nonsensical.
What other definitions of “parallel line” do you have in mind?
Congruence and equality are not the same thing. One of these axioms says that being parallel is transitive; the other says that being congruent is transitive. I agree that both notions become much less useful if transitivity does not hold, but a non-transitive congruence relation is not nonsensical.