Der Mann ist der Frau ähnlich. (the man is similar to the woman.) why does "der" preceed Frau? (instead of die) Is there something in this sentence that makes it dative?
"Danken" is a verb, "mit" a preposition, and "ähnlich" an adjective, and they all take the dative. Strictly speaking, the phrase is "ähnlich sein" - to be similar. This is the predicate, and explains why ähnlich is the last word in the sentence (non-verb parts of the predicate go to the end). It is often but not always useful to think of the dative case in German corresponding with English "to the" etc.