In a parallelogram ABCD, we are given that AD = 10 cm, BG = 12 cm, and CF = 8 cm. We need to find the length of AB.

Properties of a parallelogram:
Before we proceed with solving the problem, let's review some key properties of a parallelogram:

1. Opposite sides of a parallelogram are equal in length.
2. Opposite angles of a parallelogram are equal.
3. Adjacent angles of a parallelogram are supplementary (their sum is 180 degrees).
4. The diagonals of a parallelogram bisect each other.

Solution:
Let's label the points as shown below:

A _______ B
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D _______ C

Step 1: Identify the diagonals
In a parallelogram ABCD, the diagonals are AC and BD.

Step 2: Use the diagonal property to find the length of AC
Since the diagonals of a parallelogram bisect each other, we can conclude that AD = BC and AB = CD. Therefore, we have AD = BC = 10 cm.

Step 3: Use the given information to find the length of BD
We are given that BG = 12 cm. Since the diagonals of a parallelogram bisect each other, we can conclude that GD = BG/2 = 12/2 = 6 cm. Similarly, CF = 8 cm, so FD = CF/2 = 8/2 = 4 cm.

Since the diagonals of a parallelogram bisect each other, we can conclude that BD = BC + CD = BG + FD = 12 + 4 = 16 cm.

Step 4: Use the diagonal property to find the length of AB
Since the diagonals of a parallelogram bisect each other, we can conclude that AC = BD = 16 cm.

Now, we know that AD = 10 cm and AC = 16 cm. Using the property that opposite sides of a parallelogram are equal in length, we can conclude that AB = CD = AC - AD = 16 - 10 = 6 cm.

Therefore, the length of AB is 6 cm.