Since the right cosets of H partition G, and there are two parts of this partition |G:H|=2. Since one coset is H, the other coset Hx is G-H. The same this can be said for left cosets: xH=G-H for x outside of H. Then every right coset (H or G-H) is a left coset, which implies Hx=xH for all x in G, so H is normal.