IMO C.
From the info given, AD=DC, angle BDC=BCD=2x. angle BAC = x. You need to draw a figure to understand the question properly. In addition, i have used lami's theorem to find the lengths, but i don't know if it is required in GMAT. I knew it and hence I used it. (Lami's theorem is that for any triangle, a/SinA = b/SinB = c/SinC where a,b,c are the lengths of the sides of a triangle and A,B,C are the included angles)
Statement 1.
AD=6. This means DC=6 and hence AC=12. But there is no information regarding the angles x and 2x. Hence INSUFF.
Statement 2.
x = 36 deg
Here we get that angle BDC=BCD=2*36=72. and angle BAC=36. Thus for angle ABC, angle BAC = 36, BCA = 72 and ABC = 72. But there is no information on the length. Hence INSUFF.
Combining 1 & 2, we have for triangle ABC
LENGTH (AC) = 6 , ABC=72 and BAC=36. Using lami's theorem you will get
AC/Sin ABC = BC/Sin BAC --> 12/Sin 72 = BC/Sin 36.
You will be able to find BC from this equation. SUFF.
Hence C
Thanks.