# Verify Indifference curves of two utility function to be the same

I'm being asked to verify that two utility functions, $U(x_1,x_2)$ and $V(x_1,x_2)$ have the same indifference curves and the same MRS.

There is no problem in verifying the MRS to be the same, but how can it be verified that the two utility functions have the same I.C.? Should I assign a random level of utility and set them equal?

So let $I_U(x_1,x_2)$ denote the indifference curve (of $U$) that contains $(x_1,x_2)$. That is $$I_U(x_1,x_2) = \{(x'_1,x'_2) \in X \mid U(x'_1,x'_2) = U(x_1,x_2)\}$$ Define $I_V$ in an analogous way.
So showing the indifference curves are the same amounts to showing $I_U(x_1,x_2)= I_V(x_1,x_2)$ for all $(x_1,x_2) \in X$.