Linearly polarized Balmer line emissions from supernova remnant shocks are studied taking into account the energy loss of the shock owing to the production of nonthermal particles. The polarization degree depends on the downstream temperature and the velocity difference between upstream and downstream regions. The former is derived once the line width of the broad component of the H$alpha$ emission is observed. Then, the observation of the polarization degree tells us the latter. At the same time, the estimated value of the velocity difference independently predicts adiabatic downstream temperature that is derived from Rankine-Hugoniot relations for adiabatic shocks. If the actually observed downstream temperature is lower than the adiabatic temperature, there is a missing thermal energy which is consumed for particle acceleration. It is shown that a larger energy loss rate leads to more highly polarized H$alpha$ emission. Furthermore, we find that polarized intensity ratio of H$beta$ to H$alpha$ also depends on the energy loss rate and that it is independent of uncertain quantities such as electron temperature, the effect of Lyman line trapping and our line of sight.
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