  In this work, we provide estimates of the branching ratios, direct $CP$
asymmetries and triple product asymmetries in $B_{(s)} \to (\pi\pi)(K\pi)$
decays in the perturbative QCD approach, where the $\pi\pi$ and $K\pi$
invariant mass spectra are dominated by the vector resonances $\rho(770)$ and
$K^*(892)$, respectively. Some scalar backgrounds, such as $f_0(500,980) \to
\pi\pi$ and $K^*_0(1430) \to K\pi$ are also accounted for. The $\rho(700)$ is
parametrized by the Gounaris-Sakurai function. The relativistic Breit-Wigner
formula for the $f_0(500)$ and Flatt\'e model for the $f_0(980)$ are adopted to
parameterize the time-like scalar form factors $F_S(\omega^2)$. We also use the
D.V. Bugg model to parameterize the $f_0(500)$ and compare the relevant
theoretical predictions from different models. While in the region of $K\pi$
invariant mass, the $K^*_0(1430)$ is described with the LASS lineshape and the
$K^*(892)$ is modeled by the Breit-Wigner function. We find that the decay
rates for the considered decay modes agree with currently available data within
errors. As a by-product, we extract the branching ratios of two-body decays
$B_{(s)} \to \rho(770)K^*(892)$ from the corresponding four-body decay modes
and calculate the relevant polarization fractions. Our prediction of
longitudinal polarization fraction for $B^0\to \rho(770)^0 K^*(892)^0$ decay
deviates a lot from the recent LHCb measurement, which should be resolved. It
is shown that the direct $CP$ asymmetries are large due to the sizable
interference between the tree and penguin contributions, but they are small for
the tree-dominant or penguin-dominant processes. The PQCD predictions for the
triple product asymmetries are small which are expected in the standard model,
and consistent with the current data reported by the LHCb Collaboration.Our
results can be tested by the future precise data from the LHCb and Belle II
experiments.
