Complimentary means given free as part of the service. A complimentary breakfast, complimentary drinks, complimentary upgrade.
It is hotel and airline language for what is included in the price and presented as a gift. The other sense is expressing praise: complimentary remarks, she was very complimentary. Note the spelling trap: complimentary with an i is about praise and free gifts, and complementary with an e means completing something — complementary colours, complementary skills. The two are among the most-confused pairs in English.
1adjectivecomplimentaryGiven free as part of the service.
“The hotel offers complimentary tea and coffee in every room.”
2adjectiveexpressingExpressing praise.
“The reviews were very complimentary about her first novel.”
complimentary is a quality of being free or praising.
Italian complimento comes from Latin complere, to complete — a compliment was originally the completion of a courtesy. Complement is the same word taken directly from Latin, which is why the meanings diverged and the spellings almost did not.
From compliment + ary. complementary.
Etymology from Wiktionary, CC BY-SA
Practised in Hotel & Staying Away