Re: To supplement, or complement?
When you supplement something, you add something extra. You may supplement your regular income by doing freelance work.
When you complement something (and it's so often mis-spelled as compliment that I sometimes wonder if most people realize they are different words), you have things that work well together. A dress may complement my eyes. You choose furnishings so they complement each other.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.