why is it "par" vrai ou faux and not "avec"?

Asking “why” with (or should that be of/in/for etc) choice of preposition is generally fruitless - it is just because that’s the way it is! 

English and French use prepositions differently - it is not that one is correct and the other wrong, it just is. It is one of the hardest areas in the “acquired” language to get completely familiar and comfortable with, even after many years, for native speakers of either language.

Not quite the same in English. It would most likely be “answer with”, which I think is the reason for the question. Although “answer by” might be used, it would sound unusual and “answer by using/writing/saying “ would be more common. 

Prepositions can be quite fiddly!