ACDE is a kite and ABGF is a square. AB = BC, C + C = 96°, and AC = 12cm. Calculate, with reasons: (1) the length of AE. (2) the length of GF. (3) the size of C1 and C2. (4) the value of x.

In right-angled $\triangle ACG$ (since $AD \perp CE$ and G is the intersection):

In a kite, the diagonal that is the axis of symmetry (AD) bisects the other diagonal (CE). Since G is the intersection of AD and CE, $CG = GE$.

In right-angled $\triangle AGE$ (at G):

The length of AE is $\boxed{12 \text{ cm}}$.

(2) the length of GF.

ABGF is a square. All sides of a square are equal.
From the given information and part (1), we know $AB = 6 \text{cm}$.

The length of GF is $\boxed{6 \text{ cm}}$.

(3) the size of $\hat{C_1}$ and $\hat{C_2}$.

In right-angled $\triangle ACG$ (at G):

We are given $\hat{C_1} + \hat{C_2} = 96^\circ$.

The size of $\hat{C_1}$ is $\boxed{30^\circ}$ and $\hat{C_2}$ is $\boxed{66^\circ}$.

(4) the value of $x$.

From the given information, $AC = CD = 12 \text{cm}$.
From part (1), $AE = ED = 12 \text{cm}$.
Since all four sides are equal ($AC = CD = DE = EA = 12 \text{cm}$), ACDE is a rhombus.

In a rhombus, opposite angles are equal, and adjacent angles are supplementary (add up to $180^\circ$).
We are given $\angle ACD = \hat{C_1} + \hat{C_2} = 96^\circ$.
The angle $x$ is $\angle CDE$.
Since ACDE is a rhombus, $\angle ACD$ and $\angle CDE$ are adjacent angles.

The value of $x$ is $\boxed{84^\circ}$.