  We investigate the exclusive nonleptonic decays $B\rightarrow
\chi_{c1}K(\pi)$ in the conventional perturbative QCD (PQCD) formalism. The
predictions of branching ratios and $CP$ asymmetries are given in detail. We
compare our results with available experimental data as well as predictions of
other theoretical studies existing in the literature. It seems that the
branching ratios of $B\rightarrow \chi_{c1} K$ are more consistent with data
than the earlier analyses. For the Cabibbo-suppressed $B_s$ decay, the
branching ratio can reach the order of $10^{-5}$, which would be straight
forward for experimental observations. The numerical results show that the
direct $CP$ asymmetries of the concerned decays are rather small. The
mixing-induced $CP$ asymmetry in the $B^0\rightarrow \chi_{c1}K_S$ is very
close to $\sin{2\beta}$, which suggests that this channel offer an alternative
method for measuring the Cabbibo-Kobayashi-Maskawa (CKM) angle $\beta$. The
obtained results in the present work could be tested by further experiments in
the LHCb and forthcoming Belle II.
