Supplement and complement are used as nouns and verbs. The adjectives are supplementary and complementary.

SUPPLEMENT

Supplement as a verb means add to, increase.
She supplemented her salary by freelancing as a journalist.
The lecturer gave her students a handout to supplement the textbook.

It is also used as a noun.
After a few years the author felt that he had to write a supplement to his book.
Have you read the Sunday supplement? (Here supplement means an additional section of a newspaper.)
Many athletes use herbal supplements.

COMPLEMENT

Complement means complete, harmonise with, bring to perfection.
That tie really complements your suit.
Red wine and cheese complement each other.
Watching them together on the track through the forest I saw how well they complemented each other.

The lemon taste complemented the drink

In mathematics, angles are complementary if they add up to 90 degrees and supplementary if they equal 180 degrees.

COMPLIMENT

Compliment as a verb means to praise, to express appreciation or admiration.
He complimented her on her new dress.

It is also used as a noun.
Every time someone calls me a nerd, I take it as a compliment.
The dinner was excellent. Give my compliments to the chef!