For this problem you need to understand congruent triangles.
The sides opposite angles that are equal are also equal.

Using the above theorem we know from the figure that BD = BC (opposite angles are both 2x)
Now let's look at 1.
AD = 6. Let's see if we can get a value for BC from this.
It is given that angle BAD = x
angle BDA = 180 -2x
From this we get angle ABD to be 180 - (180 -2x + x)= x
Using our theorem we get AD = BD (opposite angles are both x)
Therefore BD =6 and hence BC =6.
So 1 is sufficient.

From 2 we get the value of x but that is not enough to figure out what BC is.
We need length of one of the sides. Hence 2 is insufficient.