What is wrong with "Eu am o câine și o pisică"? Dog is masculine so it should be "un"?
Exactly, dog is masculine and the article for masculine nouns is "un".
but can also be feminine and be "o cățea" (equivalent to cyka)
That is a "female dog". From what language is "cyka"?
When do we have to use an article in front of caine or pisica ?