  In this work we study the quasi-two-body decays $B_c \to \ K^{*} h \to K \pi
h (h = D, D_s, K, \pi, \eta, \eta')$ in the perturbative QCD (PQCD) approach.
The two-meson distribution amplitudes (DAs) $\Phi^{\text{P-wave}}_{K\pi}$ are
introduced to describe the final state interactions of the K \pi pair, which
involve the time-like form factors F_{K\pi}(s) parameterized by the
relativistic Breit-Wigner function and the Gegenbauer polynomials. We calculate
the branching ratios for these quasi-two-body decays, from which one can obtain
the branching raios for the corresponding two-body decays under the narrow
width approximation relation. We find that $B^+_c\to K^{*+}D^0$ and $B^+_c\to
K^{*0}D^+$ have the largest branching ratios, which can reach up to $10^{-6}$,
while the branching ratios for other two-body decays are very small and only
about $10^{-8}\sim10^{-7}$. As we expected that the branching ratios of the
pure annihilation decays are usually small, while in our considered such type
of decays, the channel $B_c^+ \to \bar K^{*0}K^{+}$ has the largest branching
ratio, which is near $10^{-6}$. These results are consistent with the
previously PQCD calculations obtained in the two-body framework, which can be
tested by the future LHCb experiments. For the decays $B_c^+ \to K^{*+}
D^{0}\to K^{0}\pi^+D^{0} , B_c^+ \to K^{*0}D^{+}\to K^{+}\pi^-D^{+}$ and $B_c^+
\to \bar K^{*0}D_s^{+}\to K^{-}\pi^+D_s^{+}$, we calculate their direct CP
violations and find that $A_{CP}(B_c^+ \to K^{*+}D^{0}\to
K^{0}\pi^+D^{0})=(-14.6_{-1.12}^{+9.19})\%$ is the largest one, which is
possible measured by the present LHCb experiments. For the pure annihilation
type decays, there is no CP violations because only the tree operators are
involved. Furthermore, we also give the differential distributions of the
branching ratios and the direct CP violations for the decays $B_c\to K^*
D_{(s)}\to K \pi D_{(s)}$.
