Playfair's axiom states:
Through any point in the plane, there is at most one straight line parallel to a given straight line.
This axiom is equivalent to the parallel postulate.
Intuitively, we know that if the point $P$ does not lie on the given straight line $\mathcal{l}$, there is exactly one straight line through $P$ that is parallel to $\mathcal{l}$. But how do we prove this? From Playfair's axiom, we know that there is at most one such straight line. But how do we conclude, from the first four postulates, that there is at least one?
Note: My question is different from Parallel postulate from Playfair's axiom. I have provided an answer to my own question.
Let $O$ be any point on $\mathcal{l}$. Join $OP$. Now draw a line $\mathcal{m}$ through $P$ such that the alternate interior angles, where $OP$ is a transversal to $\mathcal{l}$ and $\mathcal{m}$, are equal. Then, by Proposition 1.27 of Euclid's Elements, which does not rely on the parallel postulate, $m\parallel l$. So, there is at least one such line.