To find the converse of the original conditional statement "If this month is September, then next month ends in the letter r," we switch the hypothesis and the conclusion.
The converse is: "If next month ends in the letter r, then this month is September."
Now let's determine whether this converse is true or false.
The statement "next month ends in the letter r" is true for the months of September (October), October (November), and December (January). Thus, if next month ends in the letter r, it doesn't necessarily mean that the current month is September; it could be October or December as well.
Hence, the converse is false because there are multiple months that could have next month ending in 'r'.
A counterexample is next month being October while this month is September (which satisfies the original statement) but also could have been October (where next month would be November, which does not end in 'r').
Thus, the correct answer regarding the converse is: C - If next month ends in the letter r, then this month is September. The converse is false; a counterexample is December.