  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{align*}&\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{align*} 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.
