Angles and Parallels
In Euclid, parallelism is defined negatively by the absence of an intersection. But, even if they are not explicitly mentioned in this definition, the phenomenon of parallelism is ultimately about angles, because line intersection is also ultimately about angles (in particular, about the angle thus determined) – this is common sense, I think –; thus, in the end, parallelism is not about the angle between the parallel lines themselves (which does not exist) but about the angles produced by other, non-parallel, lines (otherwise, as I just mentioned, there would be no angles). In this context, we can think of axiom SAS as an extreme, simpler case of how to answer a more general question: how are angles are parallels related? The other extreme is to say that they are completely independent, which is what José Gil is currently exploring. But there is an ample space between one extreme and the other. Remember that, according to SAS, if two straight lines (well, to be historically accura...