These two words are pronounced identically, which is exactly why they're so easy to mix up in writing.
To complete or go well together with something.
A polite expression of praise or admiration.
The two words differ by one letter and sound the same, but they mean different things. A complement completes something or goes well with it: the sauce is a good complement to the fish. A compliment is praise: she paid him a compliment on his speech. Both work as verbs too: the colors complement each other, and the manager complimented the team.
The adjectives follow the same split, which is where many mistakes happen. Complementary means completing or going well together: complementary colors, complementary skills. Complimentary means either praising (a complimentary review) or free of charge (a complimentary breakfast), which comes from the idea of a gift given as a courtesy.
The memory trick is in the spelling. Complement has an e, like complete. Compliment has an i, and I like to receive them.
The wine is a perfect compliment to the steak.
✓The wine is a perfect complement to the steak.
The wine completes the meal; it isn't praising the steak. Complement, with an e.
She complemented my presentation after the meeting.
✓She complimented my presentation after the meeting.
She praised it, so compliment, with an i.
The hotel offers a complementary breakfast.
✓The hotel offers a complimentary breakfast.
Free of charge is complimentary, with an i.
Quick way to remember
"Complement" has an "e," like "complete" — think of it as things completing each other. "Compliment" has an "i," like "I" — think of it as something nice said to you.
Is it "a full complement of staff"?
Yes, with an e. A full complement means the complete number needed, which is the completing sense of the word.
Which is it for angles in geometry?
Complementary angles, with an e: two angles that add up to 90 degrees, completing a right angle.