I think of these in terms of "short side", "long side" and "hypotenuse." The proportions will be true that equate the ratios of corresponding sides.
I is (long side of CDB)/(short side of CDB) = (long side of BDA)/(short side of BDA)
These are corresponding sides, so the proportion is true.
II is (hypotenuse of CBA)/(long side of CBA) = (long side of CBA)/(hypotenuse of CBA)
This proportion equates a value to its inverse, so is false (since the triangle has non-zero area)
III is (long side of CBA)/(short side of CBA) = (hypotenuse of CBD)/(long side of CBD)
The sides involved here are not corresponding, so this is false.
IV is (long side of CDB)/(hypotenuse of CDB)/(long side of BDA)/(hypotenuse of BDA)
These are corresponding sides, so the proportion is true.
The appropriate choice is ...
... a. I and IV only
