While proofreading I often see the author’s confusion concerning the correct use of complement and compliment, or complementary and complimentary. Compliment is often used where complement is intended.
Complement is something that is required for completeness or harmony, or to match something, as in ‘Her hat complemented her dress’. A tip to remember the spelling is that complement and complete are spelled similarly.
Compliment is a verbal expression of admiration, such as in ‘She complimented her neighbour on his neatly trimmed and well-maintained lawn’. Following on from this complimentary tickets are free or as a compliment to someone.
Oxford Dictionaries have published a very useful blog on this subject at the following link:
http://blog.oxforddictionaries.com/2011/04/compliment-or-complement/