  In this paper, we investigate the quasi-two-body decays $B_c \to
K_0^{*}(1430,1950) D_{(s)} \to K \pi D_{(s)}$ within the perterbative QCD
(PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are
introduced to describe the final state interactions of the $K\pi$ pair, which
involve the time-like form factors and the Gegenbauer polynomials. In the
calculations, we adopt two kinds of parameterization schemes to describe the
time-like form factors: One is the relativistic Breit-Wigner (RBW) formula,
which is usually more siutable for the narrow resonances, and the other is the
LASS line shape proposed by the LASS Collaboration, which includes both the
resonant and nonresonant components. We find that the branching ratios and the
direct CP violations for the decays $B_c \to K_0^{*}(1430) D_{(s)}$ obtained
from those of the quasi-two-body decays $B_c \to K_0^{*}(1430) D_{(s)} \to K
\pi D_{(s)}$ under the narrow width approximation (NWA) can be consistent well
with the previous PQCD results calculated in the two-body framework by assuming
that $K^*_0(1430)$ being the lowest lying $\bar q s$ state, which is so-called
scenario II (SII). We conclude that the LASS parameterization is more siutable
to describe the $K_0^{*}(1430)$ than the RBW formula, and the nonresonant
components play an important role in the branching ratios of the decays $B_c
\to K_0^{*}(1430) D_{(s)} \to K \pi D_{(s)}$. In view of the large difference
between the decay width measurements for the $K_0^{*}(1950)$ given by BaBar and
LASS collaborations, we calculate the branching ratios and the CP violations
for the quasi-two-body decays $B_c \to K_0^{*}(1950) D_{(s)} \to K \pi D_{(s)}$
by using two values, $\Gamma_{K^*_0(1950)}=0.100\pm0.04$ GeV and
$\Gamma_{K^*_0(1950)}=0.201\pm0.034$ GeV, besides the two kinds of
parameterizations for the resonance $K^*_0(1950)$.
