'Adverb' is a word that is used to modify an adjective, a verb, a clause or another adverb.
In the given sentence, the verb is 'answered' and 'rude' is the way of his answer.
Option B - 'rudely' is an adverb which describes/modifies the verb 'drives'.
On the other hand, Option A - 'rude' is an adjective which can modify a noun, but not the verb.
Therefore, the suitable option to complete the given sentence is Option B - 'rudely'.
The completed sentence, thus, becomes: He answered the teacher rudely.