Measurements of $CP$ asymmetry in charmless $B\ensuremath{\rightarrow}PV$ decays are presented, where $P$ and $V$ denote a pseudoscalar and a vector meson, respectively. Five different $B\ensuremath{\rightarrow}PV$ decays from four final states, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{\ifmmode\pm\else\textpm\fi{}}{K}^{+}{K}^{\ensuremath{-}}$ and ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}{K}^{+}{K}^{\ensuremath{-}}$ are analyzed. The measurements are based on a method that does not require full amplitude analyses, and are performed using proton-proton collision data at a center-of-mass energy of 13 TeV collected by LHCb between 2015 and 2018, corresponding to an integrated luminosity of $5.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. In the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ $P$-wave, in the region dominated by the ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\ensuremath{\rho}(770{)}^{0}{K}^{\ifmmode\pm\else\textpm\fi{}}$ decay, a $CP$ asymmetry of ${A}_{CP}=+0.150\ifmmode\pm\else\textpm\fi{}0.019\ifmmode\pm\else\textpm\fi{}0.011$ is measured, where the first uncertainty is statistical and the second is systematic. This is the first observation of $CP$ violation in this process. For the other four decay channels, in regions dominated by the ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\ensuremath{\rho}(770{)}^{0}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\stackrel{(\ifmmode\bar\else\textasciimacron\fi{}\text{ })}{K}{\text{ }}^{*}(892{)}^{0}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\stackrel{(\ifmmode\bar\else\textasciimacron\fi{}\text{ })}{K}{\text{ }}^{*}(892{)}^{0}{K}^{\ifmmode\pm\else\textpm\fi{}}$ and ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\ensuremath{\phi}(1020){K}^{\ifmmode\pm\else\textpm\fi{}}$ decays, $CP$ asymmetries in the $P$-wave compatible with zero are measured.