Proof that an isometry preserves straight lines

Question

I have been trying to understand how to prove the corollary of this theorem here, for the theorem I basically prove from a point $X$ on segment $PQ$ that $F(X)$ lies on segment $F(P) F(Q)$ from the fact that isometries are distance preserving, then segment $PQ$ is contained in segment $F(P) F(Q)$. How could the same be shown for a line thus proving the corollary too?