  In this work, we investigate the quasi-two-body decays $B_c\to D^*h\to D\pi
h$ with $h = (K^0,\pi^0,\eta,\eta^{\prime})$ using the perturbative QCD(PQCD)
approach. The description of final state interactions between the $D\pi$ pair
is achieved through the two-meson distribution amplitudes(DAs), which are
normalized to the time-like form factor. The PQCD predictions on the branching
ratios of the quasi-two-body decays $B_c\to D^*h\to D\pi h$ show an obvious
hierarchy: $Br(B_{c}^+ \to D^{*+} K^{0}\to
D^0\pi^+K^{0})=({5.22}_{-0.74}^{+0.86})\times{10}^{-6}, Br(B_{c}^+ \to D^{*+}
\pi^{0}\to D^0\pi^+\pi^{0})=(0.93\pm0.26)\times{10}^{-7}, Br(B_{c}^+ \to D^{*+}
\eta\to D^0\pi^+\eta) =({2.83}_{-0.52}^{+0.59})\times{10}^{-8}$ and $Br(B_{c}^+
\to D^{*+} \eta^\prime\to
D^0\pi^+\eta^\prime)=({1.89}_{-0.36}^{+0.40})\times{10}^{-8}$.
  From the invariant mass $m_{D\pi}$-dependence of the decay spectrum for each
channel, one can find that the branching fraction is concentrated in a narrow
region around the $D^{*}$ pole mass. So one can obtain the branching ratios for
the corresponding two-body decays $B_c\to D^{*+}h$ under the narrow width
approximation. We find that the branching ratios of the decays $B_c\to D^{*+}h$
are consistent well with the previous PQCD calculations within errors. These
predictions will be tested by the future experiments.
