Prove that given $r \parallel s$, if t intersects r then it must intersect s
Solution 1:
I must assume that two lines "intersect" when they have a point in common and they don't coincide, otherwise the claim is falsified by the case $t=r$.
As per your work, using the fact that parallelism is a transitive relation defeats the purpose of disproving the assertion that it isn't transitive (which is essentially your task).
If $r\parallel s$, $A\in r\cap t$ and $t\parallel s$, then $r$ and $t$ are both parallel lines to $s$ passing through $A$. By Euclid V, this implies $r=t$.