We’ve all been scolded for using “compliment” where we should use “complement.” To compliment is to express admiration; to complement is to complete or enhance.

With a sultry smile, Hector complimented Ralph’s luxuriant eyebrows.

“That hat really complements your luxuriant eyebrows,” Hector said with a sultry smile.

I suspect this confusion is why a lot of writers shy away from “complimentary.” But, for the record, the following is correct.

“This hotel is fantastic!” Devlin gushed. “They even provide complimentary mustache wax!”

Here’s how to remember this one: the complimentary item is provided “with the compliments of the management.”