In the figure given below, $\Delta ABC\cong \Delta MBC.$ The corresponding side to $BA$ is ......

Given $△ABC\cong △MBC$.

Then the corresponding parts will also be equal.

That is by CPCT rule, corresponding parts of congruent triangles are equal.

Then, $AB=MB$,  $BC=BC$, $AC=MC$, $\angle A=\angle M$, $\angle B=\angle B$ and $\angle C=\angle C$.

Thus, $AB=MB$ $⟹$ $BA=BM$.

Hence, option $A$ is correct.