ideals in $C^*$ algebra

Question

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$.
I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact about closed ideals: Let $a\in A$ and $b\in I$ be such that $0\leq a^*a \leq b$, then $a\in I$, i.e $I$ is hereditary.
If I want to use this fact, then it seems trivial to let $b=a^*a$ as $0\leq a^*a \leq a^*a$.
Any suggestion or idea will be appreciated. Thank you