Prove that the two lines which are both parallel to the same line, are parallel to each other.

Given Three lines l, m and n in a plane such that

l ll n and m ll n.

To prove l ll m

Proof If possible, let l is not parallel to m. Then, l and m should intersect at a unique point, say P.

Thus, through a point P, outside n, there are two lines l and m, both parallel to n.

This contradicts the parallel lines axiom.

So, our assumption was wrong.

Hence, l ll m