In a normed space \(X\), by the Hahn-Banach theorem, a vector \(\xi\) is orthogonal to a vector \(\eta\) in the Birkhoff-James sense (that is, \(\|\xi+\lambda \eta \|\geq \| \xi\|\) for all \(\lambda \in \mathbb{C}\)) if and only if there is a norm one linear functional \(f\) on \(X\) such that \(f(\xi) = \|\xi\|\) and \(f(\eta) = 0\). Inspired by this fact, the authors characterize the Birkhoff-James orthogonality for elements of a Hilbert \(C^{\ast }\)-module in terms of states of the underlying \(C^{\ast }\)-algebra. They also characterize Hilbert modules in which the Birkhoff-James orthogonality coincides with the orthogonality with respect to the \(C^{\ast }\)-valued inner product by showing that it is the case only in Hilbert spaces. They characterize the norm triangle equality in terms of the Birkhoff-James orthogonality in Hilbert \(C^{\ast }\)-modules. They first show that the triangle equality \(\|\xi+\eta\|=\|\xi\|+\|\eta\|\) holds for two elements of a normed linear space exactly when \(\xi\) is Birkhoff-James-orthogonal to \(\|\eta\|\xi-\|\xi\|\eta\). They then show that the triangle equality \(\|x+y\|=\|x\|+\|y\|\) holds for elements \(x, y\) of a Hilbert \(C^{\ast }\)-module exactly when equalities in the triangle inequality \(\|\langle x,x\rangle+\langle x,y\rangle\| \leq \|\langle x,x\rangle\|+\|\langle x,y\rangle\|\) and the Cauchy-Schwarz inequality \(\|\langle x,y\rangle\|^2 \leq \|\langle x,x\rangle\|\,\|\langle y,y\rangle\|\) are attained. Related to the Birkhoff-James orthogonality and the norm-parallel relation \(||\) (\(x||y\) means \(\|\xi+\lambda \eta\|=\|\xi\|+\|\eta\|\) for some unit \(\lambda \in \mathbb{C}\)), they consider a special class of elementary operators acting on a \(C^{\ast }\)-algebra which contains the ideal of all compact Hilbert space operators.