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In this work we study the quasi-two-body decays $B_c \to \ K^{*} h \to K \pih (h = D, D_s, K, \pi, \eta, \eta')$ in the perturbative QCD (PQCD) approach.The two-meson distribution amplitudes (DAs) $\Phi^{\text{P-wave}}_{K\pi}$ areintroduced to describe the final state interactions of the K \pi pair, whichinvolve the time-like form factors F_{K\pi}(s) parameterized by therelativistic Breit-Wigner function and the Gegenbauer polynomials. We calculatethe branching ratios for these quasi-two-body decays, from which one can obtainthe branching raios for the corresponding two-body decays under the narrowwidth approximation relation. We find that $B^+_c\to K^{*+}D^0$ and $B^+_c\toK^{*0}D^+$ 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 onlyabout $10^{-8}\sim10^{-7}$. As we expected that the branching ratios of thepure annihilation decays are usually small, while in our considered such typeof decays, the channel $B_c^+ \to \bar K^{*0}K^{+}$ has the largest branchingratio, which is near $10^{-6}$. These results are consistent with thepreviously PQCD calculations obtained in the two-body framework, which can betested 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 CPviolations and find that $A_{CP}(B_c^+ \to K^{*+}D^{0}\toK^{0}\pi^+D^{0})=(-14.6_{-1.12}^{+9.19})\%$ is the largest one, which ispossible measured by the present LHCb experiments. For the pure annihilationtype decays, there is no CP violations because only the tree operators areinvolved. Furthermore, we also give the differential distributions of thebranching ratios and the direct CP violations for the decays $B_c\to K^*D_{(s)}\to K \pi D_{(s)}$.

Quasi-two-body decays $B_c \to K^{*} h \to K \pi h $ in the perturbative QCD