# Transitivity of parallel lines

I cam across a question (in my textbook) about proofs with parallel lines. The question is: Prove that the property that || is transitive implies that for any point P and line l, there is at the most 1 line through P that is parallel to the line l.

In other words the question is asking me to prove that (P is on q & P is on s & l||q & l||s) --> q = s, using the transitive property.

I know that the transitivity property tells me that l||m & m||n --> n||l, but I am not sure how to do this proof.

Any hints or solutions would be much appreciated!! :)

• The notation in your second paragraph is messed up. On the left side, you have q,s,n but on the right side you have m and n. Once you correct this, you should be able to see the solution. – Ted Sep 27 '13 at 6:49
• @Ted After correcting it, I can see that q || s, but I can't think of the next step – Michael Ferashire Silva Sep 27 '13 at 7:01
• q||s, but they both go through P, so ... – Ted Sep 27 '13 at 16:38