If \(\mathbb{D}\) denotes the the open unit disk and \(0<p<+\infty \), the Bergman space \(A^{p}\) is the space of analytic functions on \(\mathbb{D}\) such that \[ \| f\| _{p} =\left( \int_{\mathbb{D}} | f(z)| ^{p} dA(z) \right)^{\frac {1}{p}} \] is finite, where \(dA\) is the normalized area measure on \(\mathbb{D}\). A closed subspace \(I\) of \(A^{p}\) is {invariant} if \(zf\) belongs to \(I\) for any function \(f\in I\). In the present paper, the authors show that certain invariant subspaces of \(A^{p} \) are complemented in \(A^{p}\) (in the context of Hardy spaces, every invariant subspace of \(H^{p}(\mathbb{D})\), \(1<p<+\infty \), is complemented in \(H^{p}(\mathbb{D})\)). If \(Z=(z_{n})\) is an \(A^{p}\)-interpolating sequence, the invariant subspace \(I_{Z}^{p}\) which consists of all the functions in \(A^{p}\) that vanish on \(Z\) is complemented in \(A^{p}\) for \(0<p<+\infty \). The second case which the authors investigate is the case of the space \(I_{\sigma }^{p}\) generated by the singular inner function \[ S_{\sigma }(z)- \exp \left(-\sigma \frac {1+z}{1-z}\right), \] where \(\sigma >0\) and \(1<p<+\infty \), which is complemented in \(A^{p}\). The proof of this last result uses orocycles, which are the level curves of the function \(S_{\sigma }\), in order to construct a bounded projection of \(A^{p}\) onto \(I_{\sigma }^{p}\).