For me, the easiest way to remember the difference between these two is to remember the definitions of compliment (something nice you say about someone) and complement (something that completes a set or group).
Something that is complimentary is either a) free or b) flattering. (Maybe another way to remember it is to think "I like things that are complImentary". Or maybe that's just really corny.)
Something that is complementary completes a set, matches a pair, or fills out a group. Angles, base pairs, and wines can be complementary, but statements and newspapers generally aren't.
So, you can sip complimentary coffee while contemplating the complementary angles on the rafters above your head. But if you start encountering complimentary angles, you might want to get your eyes (ears?) checked...
7 years ago