Translation:We left the house and we went to the airport.
I have a similar problem. I kept hearing "a teach" and was disappointed when "We left her house..." was marked wrong. I understand that the "n" sound can frequently get dropped from the definite article in conversational Irish, but this one was particularly difficult to discern.
Is "go dtí" a different type of preposition than "ar"? Because I thought I had figured out that as an object of a preposition, you would not have the "t-" in front of a noun that starts with a vowel. But yet here it is. (The other sentence I'm thinking of in another lesson is "I eat breakfast at the airport" and the phrase is "ar an aerfort").
There isn't a single rule that applies to all prepositions. Most, but not all, of the "simple prepositions" cause eclipsis after an (ar and ag do, i and do don't). Some of them cause lenition when there isn't a definite article, some of them don't. Some prepositions combine with pronouns to create prepositional pronouns (agam, romhat, linn, etc), some, like go, don't. Some "derived prepositions" put the following preposition in the genitive case.
So unfortunately there isn't a single one-size-fits-all rule when it comes to prepositions.