"My father has a restaurant."
Translation:Mio padre ha un ristorante.
You mean family "nouns" do not use "definite article", "mio" is the possessive adjective. A pronoun would replace a noun. There are a couple of exceptions: http://italian.about.com/od/grammar/a/italian-possessive-adjectives.htm
Oddly, this source says differently. Please scroll down there to see what I mean : http://italian.about.com/od/grammar/a/italian-possessive-adjectives.htm
Duo is trying to teach the difference between the English words father and dad, and the Italian padre and papa.
father = padre dad = papa, babbo and any other familiar or diminutive terms there are for ones father.
So if the sentence was 'My dad has a restaurant', your translation would be accepted.
I know, I struggle with it too, but here are some guidelines. The statement about the lion is actually “il mio papà ha un leone” and that padre/papà is what makes the difference. The MAIN exceptions to using the definite article are:
Do NOT use a definite article (il, la, etc) in front of singular nouns referring to close family members (padre, madre, sorella, etc). HOWEVER, The definite article IS USED with nouns that are VARIATIONS of the basic forms, such as babbo, papà (dad), mamma (mom), and fratellino (little brother).
A definite article IS USED when referring to parts of the body and articles of clothing HOWEVER, the possessive adjective (mio, sua, etc) is NOT used. Ex. Mi fa male la testa. (My head hurts.) Cerco le scarpe. (I am looking for my shoes.)
A definite article is NOT USED WITH the possessive adjective when the adjetive comes after the noun in a statement of FACT or ECLAMATION! Ex. Mio dio, questo è confuso! (My god, this is confusing!)