Using either (1) or (2) we can say Line AB is parallel to CD, but line FG can be drawn anyway such that it intersects the 2 lines at point E.

It could be drawn parallel as shown in the figure, or it could not be parallel and still intersect at point E. There is no info about FG in either options.

Should be EIMO

In the circle pictured with diameter CD, segments AD and BC intersect at point E. E lies on segment FG. Are the lines FG and AB parallel to CD?

(1) Triangle CED is isosceles with base CD.

=> ∠ECD = ∠EDC and EC = ED,
=> ∠AEB = CED and QE = EB,
=> AE/EC = BE/ED, ∠AEB = ∠CED => AB || CD (Similar triangles?)

Insufficient !

(2) ∠BAD = ∠ADC => AB||CD (Alternate Interior angles)

Insufficient !

No mention of FG.

Yea, I don't think there's much more to say than that FG could be drawn in pretty much any way you want, and there is no mention of it in the statements.

This question tricks you into over thinking the rules for similar triangles - asking how many angles you need and trying to figure that out. Meanwhile the quesiton never gives you any information regarding where FG is or any information about the angles related to FG. Therefore none of the statements are sufficient and the answer must be E.