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Hydrogen bonding is actually a very specific type of dipole-dipole interaction, between and H-N, H-O, or H-F bond and the lone pair of another electronegative element. So in that sense, yes: if a molecule has hydrogen bonding, it technically has dipole-dipole interactions as well.
A hydrogen bond is a dipole-dipole force but not all dipole-dipole forces are hydrogen bonds--- only when a H bonds with a highly electronegative atom (N,O,F). Hydrogen bonds are stronger than normal dipole-dipole forces.
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