If something is untenable, you can't defend it or justify it. If your disagreement with your teacher puts you in an untenable position, you better just admit you made a mistake and get on with it.
When untenable entered English in the 17th century it meant "unable to be held against attack." That sense still holds true: you can use the adjective untenable to describe any situation, position, or theory that simply can't be defended. Untenable is a great word to use when you want to criticize something, whether it's a flawed system or a referee's bad call.