The adjective "complementary" means "serving to complete, fill out, or make perfect." Remember that the letter “e” is in both "complementary" and "complete."

The adjective "complimentary" means "favorable; expressing respect or affection; given free."

We didn't spend much money last night because we had complimentary tickets to a gallery opening.

Our tickets were free.

The gallery provided refreshments, including a variety of cheeses and complementary wines.

The wines went well with the cheeses.

The show was a success; we heard many complimentary remarks about the art.

People said nice things.

The paintings featured shapes in complementary colors. My favorite was the one with red squares and green circles, but I also liked the one with blue and orange triangles.

Colors that are complementary combine to form white or a shade of gray.