A small fraction (<10%) of the SDSS main galaxy (MG) sample has not been targeted with spectroscopy due to the effect of fiber collisions. These galaxies have been compiled into the input catalog of the LAMOST ExtraGAlactic Surveys and named the complementary galaxy sample. In this paper, we introduce this project and status of the spectroscopies associated with the complementary galaxies in the first two years of the LAMOST spectral survey (till Sep. of 2014). Moreover, we present a sample of 1102 galaxy pairs identified from the LAMOST complementary galaxies and SDSS MGs, which are defined as two members that have a projected distance smaller than 100h_70_^-1^kpc and a recessional velocity difference smaller than 500km/s. Compared with galaxy pairs that are only selected from SDSS, the LAMOST- SDSS pairs have the advantages of not being biased toward large separations and therefore act as a useful supplement in statistical studies of galaxy interaction and galaxy merging.