This is a great question, and one with a very common GMAT strategy that I espouse to all my students: just start filling in variables for all possible angles, and see what happens.
(I'm going to refer to angles just as the three letters.)
So, for example, is BDC=2x, then we know that BDA=180-2x.
Now, we take that information and apply it to what we know in the triangle to the left:
ADB + DBA + BAD = 180.
(180-2x) + DBA + x = 180
DBA = 180 - x - (180-2x) = x
If DBA= x, and BAD=x, then triangle BAD is isosceles.
Statement one says that segment AD is 6. With our knowledge of isosceles triangles, we then know that BD is also 6. And since the right- hand triangle is isosceles based on the information given, then segment BC must also be 6.
Statement one is sufficient.
Statement two just talks about angles, and gives no information about side lengths.
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