Supplementary is a little something extra to fill in a gap, like when your teacher suggests supplementary reading material that you may or may not get around to checking out.
Supplementary can be an important part of something or just extra support. The word supplement comes from the Latin supplementum for "something added to fix a deficiency" and the suffix -ary means "connected with." Put them together and you have something connected with filling in. The supplementary support of digging a moat around your sand castle might save it from a wave, but that supplementary reading on your syllabus isn't required (although it might save your grade).