If X is in NP-complete and complement(X) is in NP, show that for all Y in NP, complement(Y) is also in NP
Note that

$$
A \leq B \iff \overline{A} \leq \overline{B}
$$

where $\overline{A}$ denotes the complement of $A$. Indeed, if $f$ is a reduction for either side, it is also a reduction for the other side (exercise: prove this using the definition of $\leq$).

From this, your statement quickly follows.