In traditional logic, the only way
"If the circle is blue, then the rectangle is not green"
can be false is if the antecedent ("the circle is blue") is true, but the consequent ("the rectangle is not green") is false.
We're told that it is false, so we now know "the circle is blue" is true and "the rectangle is not green" is false. If "the rectangle is not green" is false, then "the rectangle is green" is true. So now we know two things are true for sure:
"the circle is blue"
"the rectangle is green"
We're also told something about the converse, but we don't actually need that fact to answer the only other statement you asked us to classify as true or false (I'm guessing that somehow we're missing other statements to consider). So I'm not going to talk about that bit.
We're asked to consider the truth of "the circle is blue if and only if the rectangle is green". That's and "if and only if", so it's true when the conditional
"if the circle is blue, then the rectangle is green" is true AND
"if the rectangle is green then the circle is blue" is true.
We know that "the circle is blue" is true and "the rectangle is green" is true, so each of those conditionals I just mentioned are true: each has a true antecedent (the if part) and a true consequent (the then part).
We could demonstrate this all with truth tables, too. Hopefully this helps. Of course, if you want to schedule a session with me to go over this in detail or other examples, please do!
