We study the shapes of galaxy dark matter haloes by measuring the anisotropy of the weak gravitational lensing signal around galaxies in the second Red-sequence Cluster Survey (RCS2). We determine the average shear anisotropy within the virial radius for three lens samples: all galaxies with 19<m_r<21.5, and the `red and `blue samples, whose lensing signals are dominated by massive low-redshift early-type and late-type galaxies, respectively. To study the environmental dependence of the lensing signal, we separate each lens sample into an isolated and clustered part and analyse them separately. We also measure the azimuthal dependence of the distribution of physically associated galaxies around the lens samples. We find that these satellites preferentially reside near the major axis of the lenses, and constrain the angle between the major axis of the lens and the average location of the satellites to <theta>=43.7 deg +/- 0.3 deg for the `all lenses, <theta>=41.7 deg +/- 0.5 deg for the `red lenses and <theta>=42.0 deg +/- 1.4 deg for the `blue lenses. For the `all sample, we find that the anisotropy of the galaxy-mass cross-correlation function <f-f_45>=0.23 +/- 0.12, providing weak support for the view that the average galaxy is embedded in, and preferentially aligned with, a triaxial dark matter halo. Assuming an elliptical Navarro-Frenk-White (NFW) profile, we find that the ratio of the dark matter halo ellipticity and the galaxy ellipticity f_h=e_h/e_g=1.50+1.03-1.01, which for a mean lens ellipticity of 0.25 corresponds to a projected halo ellipticity of e_h=0.38+0.26-0.25 if the halo and the lens are perfectly aligned. For isolated galaxies of the `all sample, the average shear anisotropy increases to <f-f_45>=0.51+0.26-0.25 and f_h=4.73+2.17-2.05, whilst for clustered galaxies the signal is consistent with zero. (abridged)
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