  Measurements of $CP$ asymmetry in charmless $B\to PV$ decays are presented,
where $P$ and $V$ denote a pseudoscalar and a vector meson, respectively. Five
different $B\to PV$ decays from four final states, $B^\pm \to \pi^\pm \pi^+
\pi^-$, $B^\pm \to K^\pm \pi^+ \pi^-$, $B^\pm \to K^\pm K^+ K^-$ and $B^\pm \to
\pi^\pm K^+ K^-$ 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 fb$^{-1}$. In
the $\pi^+ \pi^-$ $\textit{P}$-wave, in the region dominated by the $B^\pm \to
\rho(770)^0 K^\pm$ decay, a $CP$ asymmetry of $\mbox{$A_{CP}$ = +0.150 $\pm$
0.019 $\pm$ 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
$B^\pm \to \rho(770)^0 \pi^\pm$, $B^\pm \to \overset{(-)}{K^*}(892)^0 \pi^\pm$,
$B^\pm \to \overset{(-)}{K^*}(892)^0 K^\pm$ and $B^\pm \to \phi(1020) K^\pm$
decays, $CP$ asymmetries in the $\textit{P}$-wave compatible with zero are
measured.
