Summary: Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies \(C^*\)-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple \((P, Q, H)\) is said to be matched if \(H\) is a Hilbert \(C^*\)-module, \(P\) and \(Q\) are projections on \(H\) such that their infimum \(P \wedge Q\) exists as an element of \(\mathcal{L}(H)\), where \(\mathcal{L}(H)\) denotes the set of all adjointable operators on \(H\). The \(C^*\)-subalgebras of \(\mathcal{L}(H)\) generated by elements in \(\{P - P \wedge Q, Q - P \wedge Q, I\}\) and \(\{P, Q, P \wedge Q, I\}\) are denoted by \(i(P, Q, H)\) and \(o(P, Q, H)\), respectively. It is proved that each faithful representation \((\pi, X)\) of \(o(P, Q, H)\) can induce a faithful representation \((\widetilde{\pi}, X)\) of \(i(P, Q, H)\) such that \[ \begin{aligned} \widetilde{\pi} (P - P \wedge Q) = \pi (P) - \pi (P) \wedge \pi (Q),\\ \widetilde{\pi} (Q - P \wedge Q) = \pi (Q) - \pi (P) \wedge \pi (Q). \end{aligned} \] When \((P, Q)\) is semi-harmonious, that is, \(\overline{\mathcal{R}(P + Q)}\) and \(\overline{\mathcal{R}(2I - P - Q)}\) are both orthogonally complemented in \(H\), it is shown that \(i(P, Q, H)\) and \(i(I - Q, I - P, H)\) are unitarily equivalent via a unitary operator in \(\mathcal{L}(H)\). A counterexample is constructed, which shows that the same may be not true when \((P, Q)\) fails to be semi-harmonious. Likewise, a counterexample is constructed such that \((P, Q)\) is semi-harmonious, whereas \((P, I - Q)\) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert \(C^*\)-modules are also provided.