A complement is something that makes up a satisfying whole with something else. Those shiny shoes you just bought form a perfect complement to your shiny hat.
Complement is a good word with the bad luck to sound exactly like the more common compliment. While both words come from the Latin complementum, "something that fills up or completes," complement keeps both the -e- and the meaning. It's also a verb; if you and your partner complement each other, you make a perfect pair. This is very different from complimenting each other by praising each other's outfits.
