  Two-body charmless hadronic $B_c$ decays involving a light
$1^3\!P_2$-tensor($T$) meson are investigated for the first time within the
framework of perturbative QCD(pQCD) at leading order, in which the other meson
is the lightest pseudoscalar($P$) or vector($V$) state. The concerned processes
can only occur through the pure weak annihilation topology in the standard
model. We predict the CP-averaged branching ratios and polarization fractions
of those considered decays in Cabibbo-Kobayashi-Maskawa(CKM) favored and
suppressed modes. Phenomenologically, several modes$-$such as the $B_c \to
K_2^*(1430) K$ and the CKM-favored $B_c \to TV-$have large decay rates of
$10^{-6}$, which are expected to be detected at Large Hadron Collider
experiments in the near future. Moreover, all of the $B_c \to TV$ modes are
governed by the longitudinal amplitudes in the pQCD calculations and the
corresponding fractions vary around $78\%$-$98\%$. A confirmation of these
results could prove the reliability of the pQCD approach used here and further
shed some light on the annihilation decay mechanism.
