What is the minimum number of different types of tRNA molecules that must exist in a cell?

there must be at least 60 because a codon consists of three base pairs and there are 20 amino acids

There must be at least 20 -- one for each kind of amino acid.  As it turns out, there's about 40.  How, you may ask, can that be, when there's 61 different codons?  The anticodons on some tRNA have inositol or other modified nitrogenous bases, which don't quite follow Chargaff's rules for base pairing, so one tRNA anticodon can actually be matched with more than one mRNA codon.> What part of translation depends on the same basepairing rule that is used in transcription and dna replication?

Your teacher wants you to answer:  the matching of the tRNA anticodon with the mRNA codon.
As I mentioned above, that's not quite the case, since tRNA can contain inositol or other modified nitrogenous bases that don't follow the same base pairing rules as you have in transcription and replication.