This is another post written because of suggestions I have received from readers of my blog (thank you).

My compliments to the chef.

Compliment:

• An expression of admiration or praise.
• Compliments are formal greetings.

Complement:

• A person or thing that completes something.
• An accompaniment to something that makes it perfect or brings out its good qualities.
• A full number or complete amount of something.
• The full amount of officers and crew needed to man a ship.
• A word or words added to the verb to complete the meaning of the predicate in a sentence (grammar).
• The angle added to a specified angle to produce a right angle (geometry).
• The members of a set that are not found in a given subset (maths).
• A group of proteins found in blood plasma and tissue fluid that combine with an antigen-antibody complex to cause the breakdown of foreign cells (physiology).

The meaning is the same when using the words as verbs. You compliment me on my dress. My shoes complement my dress.

Complimentary and complementary

The difference here is largely the same as above. However, complimentary has an additional meaning to expressing admiration or praise: it can also mean given free of charge.

Complementary means completing or forming a complement, or refers to the relationship between things that go well together.

Spelling

It may help as an aid to spelling to remember that a complement completes something and complete is spelt with an e (as well as the ‘e’ sound when pronounced).