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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 $AB$ ?

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    I deleted the tag [tag:transversality] as it is irrelevant to your question. Even your question deals with some kind of transversality, this precise tag is for transversality in differential topology. Please next time read the tag excerpts, they are here as a guidance to use them. – 2017-01-31

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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.

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    In simpler terms, it is not possible to achieve $180^{\circ}$ unless every point on both parallel lines are touching each other (i.e. there is only one line). That is why $180^{\circ}$ only forms on a straight line. – 2017-01-28