Explain how the root "sist" contributes to the meaning of these words: Consistent Resistance Subsist
The root "sist" means "stand," so the words on your list all have something to do with "standing."
A person who is consistent in his/her beliefs or actions is one who "constantly adheres to the same principles or is not self-contradictory" according to Dictionary.com. Consistent can also mean "holding firmly together," so a consistent person is one who takes a firm stand and does not falter from it.
Resistance has several meanings, the first being, "the act or power of opposing or withstanding." Again, a person who resists, stands strong in his/her beliefs or actions. The word "resistance" is also used as the name of an underground group of people who work in opposition to a government or other power with whom they disagree. They take a stand and are willing to do whatever it takes to overtake that power.
Last, "subsist" can mean "to remain alive or to endure." When one endures something, he/she stays standing in spite of the conditions surrounding him/her.