In the figure above, if x and y are each less than 90 and PS

[RonPurewal]

rachel, no. the question specifically asks whether one segment is less than another segment -- not just whether the segments are equal or not equal.
if the problem statement said "are lengths pq and sr equal?" then your approach would be valid. with the question as stated, it isn't.

[lmaura429]

This problem confuses me because isn't the diagram 2 parallel lines cut by a transversal, and therefore, shouldn't X = Y? Once I hit statement 1, I knew that I had something wrong on that front but my approach was:

(1) X > Y. The SR angle with y (given) is equal to angle PSR because the alternate interior angles are equal. Therefore, if X > Y, SR is opposite of X and PQ is opposite of Y, so SR > PQ. Sufficient.

(2) Insufficient. Doesn't tell me anything of use.

Was my logic correct?

[RonPurewal]

This problem confuses me because isn't the diagram 2 parallel lines cut by a transversal, and therefore, shouldn't X = Y?

no. wrong parallel lines.
for x and y to be the same, the two diagonal lines would have to be parallel. but what you're given is that the two horizontal lines are parallel.
if this is not crystal clear, take a red pen or highlighter and trace out the lines that form angles x and y. then take a look at which pair of (supposedly) parallel lines you've highlighted.

(1) X > Y. The SR angle with y (given) is equal to angle PSR because the alternate interior angles are equal. Therefore, if X > Y, SR is opposite of X and PQ is opposite of Y, so SR > PQ. Sufficient.

don't know exactly what you mean by "opposite" here.
if you are trying to use the "longer length is found across from larger angle" rule, then, no (although you get really, profoundly, don't-bother-playing-the-lottery-anymore lucky in this one particular problem). that rule only applies to triangles, not to any random shape containing angles and lines.
(if the angles in this figure were greater than 90º, then the inequalities would work out in exactly the opposite way.)

if you meant something else, go ahead and explain. thanks.