My book says relation "parallel" on the set of lines in the plane not transitive.
And the definition in the book given is :
A relation $R$ on a set $A$ is transitive if whenever $aRb$ and $bRc$ then $aRc$, that is if whenever $(a,b),\,(b,c)\in R$ then $(a,c)\in R$.
What I thought is that if $a\parallel b$ and $b \parallel c$ then $a \parallel c$. Therefore it is transitive.