Please see the following pictures
In the image there are two parallel lines(blue coloured) and there is a transversal segment $AB$. In the image, you can see it clearly that as we increase the size of the transversal $AB$, the measure of $\angle{BAC} $ also increases.
So,my question is that would a time come when $\angle{BAC} =180^\circ $ if we go on increasing the size of $AB$ ?
No, since there is no "infinitely far away". Wherever you put $B$, $\triangle ABC$ is a triangle with positive area (the area actually stays the same no matter where you put $B$: it will always be $|AC|$ times half the distance between the blue lines), and therefore it is non-degenerate, which means all angles are strictly positive. But any angle between $0^\circ$ and $180^\circ$ is possible to get. $179.999999^\circ$? No problem, just put $B$ far enough away. How about $(180 - 10^{-1000000})^\circ$? Also very much possible.