by **RonPurewal** Wed Nov 12, 2008 8:16 am

saurav's solution is good once again.

i'm partial to explanations with more words than algebraic expressions, though, so here's an equivalent version that has more words in it.

statement (1)

we know that the angles are 90Â°, 90Â°, x, y, and that all of them sum to 360Â°.

by subtraction, x + y must be 180Â°.

it's possible that one of them could be 60Â° (if the other one is 120Â°), but it's also possible for neither of them to be 60Â° (if they have any measures other than 60Â° and 120Â°).

insufficient.

statement (2)

you could pick any 1:2 ratio, from 0.0000001Â° and 0.0000002Â° all the way up to 89.99999Â° and 179.99998Â°, and just select the remaining 2 angles so that the sum of all four of them is 360Â°.

this includes possibilities in which one of the angles is 60Â°, as well as possibilities in which none of the angles has that measure.

insufficient.

(together)

the trap here is to assume that the angles are 90Â°, 90Â°, xÂ°, and 2xÂ°. that's one possibility, but not the only one. in this case, x = 60 and 2x = 120.

however, it's possible that the 90Â° angle is the "2x" in this problem. this would mean that a third angle was 45Â° (so that 90Â° and 45Â° provide the required 2:1 ratio), and, by subtraction, the last angle is 135Â°.

therefore, 60Â° could be either present or absent.

still insufficient.

answer (e).