Dijet angular distributions are measured over a wide range of dijet invariant masses in $pp$ collisions at $\sqrt{s}=7\text{ }\text{ }\mathrm{TeV}$, at the CERN LHC. The event sample, recorded with the CMS detector, corresponds to an integrated luminosity of $36\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$. The data are found to be in good agreement with the predictions of perturbative QCD, and yield no evidence of quark compositeness. With a modified frequentist approach, a lower limit on the contact interaction scale for left-handed quarks of ${\ensuremath{\Lambda}}^{+}=5.6\text{ }\text{ }\mathrm{TeV}$ (${\ensuremath{\Lambda}}^{\ensuremath{-}}=6.7\text{ }\text{ }\mathrm{TeV}$) for destructive (constructive) interference is obtained at the 95% confidence level.