Talk:Parallelism is Transitive

So something came to mind when I was doing this: can we prove that a transversal that cuts one of two parallel lines also cuts another? It's obviously true, but I don't immediately see how to prove it. I'll come back to it at some point, but fell free to do it/post any ideas here. --Cynic (talk) 02:59, 2 June 2009 (UTC)

Quick thought: can't you use the fact that a transversal is not parallel to a given line, so therefore is not parallel to the one parallel to it, and therefore by 5th Postulate cuts the parallel? Or does that require circularity? I think it's going to require the 5th postulate anyway because in non-Euclidean geometry it's not generally true. --Matt Westwood 17:49, 2 June 2009 (UTC)

Oh, it's obvious now. Both the parallel (middle) line and the transversal go through a single point, and since there's only one line parallel to a given line through a point not on it, and we know that the parallel is parallel, the transversal must intersect it. I'll think about how to phrase that well and add it in. --Cynic (talk) 19:17, 2 June 2009 (UTC)