complimentary (adj.)

"act or expression of civility, respect, or regard" (or, as Johnson defines it, "An act, or expression of civility, usually understood to include some hypocrisy, and to mean less than it declares"), 1570s, complement, ultimately from Latin complementum "that which fills up or completes" (see complement, which is essentially the same word), the notion being "that which completes the obligations of politeness."

The spelling of this derived sense shifted in English after c. 1650 to compliment, via French compliment (17c.), which is from Italian complimento "expression of respect and civility," from complire "to fill up, finish, suit, compliment," from Vulgar Latin *complire, for Latin complere "to complete" (see complete (adj.)).

By early 19c. the meaning had been extended to "an expression of praise or admiration. Meaning "a present or favor bestowed, a complimentary gift" is from 1722.

adjective and noun word-forming element, in most cases from Latin -arius, -aria, -arium "connected with, pertaining to; the man engaged in," from PIE relational adjective suffix *-yo- "of or belonging to." The neuter of the adjectives in Latin also were often used as nouns (solarium "sundial," vivarium, honorarium, etc.). It appears in words borrowed from Latin in Middle English. In later borrowings from Latin to French, it became -aire and passed into Middle English as -arie, subsequently -ary.

"complimentary ticket," 1885, short for complimentary. Meaning "nonpaying guest" is attested by 1930s; generalized to "anything given free" by 1960s. As a verb, by 1974. Related: Comped; comping. As a shortening of compensation (especially worker's/workman's) it was in use by 1970s.