Hi All,

We're told that each side of parallelogram P has length 1. We're asked for the area of P. This is a great 'concept question', meaning that if you know the concepts involved, then you don't actually have to do much math to get the correct answer. Here, since we know all 4 sides of the parallelogram, if we have ANY of the 4 angles, then can determine the area.

(1) One angle of P measures 45 degrees.

A parallelogram is 360 degrees and 'opposite' angles are equal. With the information in Fact 1, we know that the 4 angles are 45/135/45/135. Combined with the side lengths (which we know are all 1s), we can determine the exact area of this shape.
Fact 1 is SUFFICIENT

(2) The altitude of P is √2/2

The information in Fact 2 requires a bit more work, but we can now draw 2 RIGHT triangles "inside" the parallelogram. Each of those right triangles would have a hypotenuse of 1 (since the side lengths are 1s) and a 'height' of √2/2. We could determine the exact value of the 3rd side and the two angles (it ends up being a 45/45/90 right triangle), so we can then determine the area of the overall shape.
Fact 2 is SUFFICIENT

Final Answer: D

GMAT assassins aren't born, they're made,
Rich