In order to prove that triangles are congruent, all the angles and sides have to be congruent. What if we aren't given any angles? We can use the SSS postulate (which has no A's—unlike your geometry tests). If all the sides are congruent, then the two triangles are congruent.
So are all the sides congruent? We're given that AB ≅ BC ≅ AD ≅ CD. Two of the sides match, but that isn't enough to prove congruence. What about the third side, AC? It's a side for both ∆ABC and ∆ADC, and it' congruent to itself, for sure. That means all three sides are congruent to one another, meaning ∆ABC ≅ ∆ADC.