Complement(ary) vs. Compliment(ary)

The verb complement means “to complete; to fill out by supplying what something else lacks.” Remember this by relating it to the use of the term complementary in math: it describes two acute angles that together form a right angle.

They were a good team because her technical skills complimented his creativity.

They were a good team because her technical skills complemented his creativity.
They were a good team because their talents were complementary.

One meaning of complimentary is “free of charge”:

complimentary beverages on the flight

A compliment is an expression of praise; as a verb, to compliment means “to express praise.” Therefore, complimentary can also mean “expressing praise”:

Give my compliments to the chef.
highly complimentary review of the restaurant