Question

are theorems in a mathematical system proved by deductive reasoning?

Answer 1

Sometimes. There are some other methods of proof. Contradiction and induction come to mind.

More formally, a deduction concludes with a modus ponens (which agrees with the rest of the proof), or in formal language:
If P then Q
P->Q
A contradiction concludes with a modus tollens (the opposite of the assumption), or in formal language:
If P then Q
not Q -> not P
Both are solid, reliable ways of prove something.
Induction is more subtle and harder to think about. It revolves around thinking of a simple base case and then assuming the solution has a certain form T. Then you have to proof that T(n+1) is valid.
The wikipedia may give some insight to this: http://en.wikipedia.org/wiki/Mathematical_induction

Answer 2

Yes i know that. But according to my school, deductive is what they wanted.

Answer 3

:-)