How do I prove that if $A$ and $B$ are complementary then $A^\perp$ and $B^\perp$ are also complementary in $\mathcal{H}$ Hilbert space?

Question

I have the following problem: I know that $A$ and $B$ are two closed complementary subspaces in an Hilbert space $\mathcal{H}$, i.e. $\mathcal{H}=A\oplus B$ (where $\oplus$ is the direct sum of two closed subspaces). I have to prove that $A^\perp$ and $B^\perp$ are also complementary.
For now, I only managed to prove that $\overline{A^\perp\oplus B^\perp}=\mathcal{H}$ using the identity $(R\oplus S)^{\perp}=R^{\perp}\cap S^{\perp}$ and plugging in $R=A^{\perp}, S=B^\perp$ and taking into account that $A$ and $B$ are closed to say $(A^\perp)^\perp=A$, $(B^\perp)^\perp=B$.
I know that in general the direct sum of two closed subspaces is not closed, so I cannot conclude that $\overline{A^\perp\oplus B^\perp}=A^\perp\oplus B^\perp$. If someone can give me an hint on how to solve this it would be greatly appreciated.