  Extensive air showers, originating from ultra-high energy cosmic rays, have
been successfully measured through the use of arrays of water-Cherenkov
detectors (WCDs). Sophisticated analyses exploiting WCD data have made it
possible to demonstrate that shower simulations, based on different
hadronic-interaction models, cannot reproduce the observed number of muons at
the ground. The accurate knowledge of the WCD response to muons is paramount in
establishing the exact level of this discrepancy. In this work, we report on a
study of the response of a WCD of the Pierre Auger Observatory to atmospheric
muons performed with a hodoscope made of resistive plate chambers (RPCs),
enabling us to select and reconstruct nearly 600 thousand single muon
trajectories with zenith angles ranging from 0$^\circ$ to 55$^\circ$.
Comparison of distributions of key observables between the hodoscope data and
the predictions of dedicated simulations allows us to demonstrate the accuracy
of the latter at a level of 2%. As the WCD calibration is based on its response
to atmospheric muons, the hodoscope data are also exploited to show the
long-term stability of the procedure.
