We introduce the notion of a separated pair of closed submodules within the framework of Hilbert $C^*$-modules. We demonstrate that even in the context of Hilbert spaces, this idea has several insightful characterizations enriching the understanding of separated pairs of subspaces in Hilbert spaces. Let $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr H$ and $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr K$ be orthogonally complemented closed submodules of a Hilbert $C^*$-module $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr E$. We establish that $ (\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr H,\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr K)$ forms a separated pair in $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr{E}$ if and only if there exist idempotents $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1$ and $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2$ such that $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1=0$, $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr R(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1)=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr H$, and $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr R(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2)=\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr K$. We show that $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr R(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1+\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\lambda\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2)$ is closed for each $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\lambda\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\in \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathbb{C}$ if and only if $\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr R(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_1+\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Pi_2)$ is closed. We utilize the localization of Hilbert $C^*$-modules to define the angle between closed submodules. We prove that if $(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr H^\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp,\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr K^\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp)$ is concordant, then $(\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr H^{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp},\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\mathscr K^{\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\perp})$ is a separated pair if the cosine of this angle is less than one. We also present some surprising examples to illustrate our results.

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