To find the area of parallelgram ABCD, draw perpendicular segment, BH, as shown. Since BH is the side opposite a `45^(@)` angle in a right triangle.

Answer

To find the area of parallelogram ABCD, we will follow these steps:

### Step 1: Identify the height and base of the parallelogram.
We have a perpendicular segment BH drawn from point B to line AD. This segment represents the height (H) of the parallelogram. The base (B) of the parallelogram can be considered as the length of side AD (or BC, since opposite sides of a parallelogram are equal).

### Step 2: Determine the height using the sine function.
In triangle ABH, we know that angle ABH is 45 degrees. The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, we can write:

\[ \sin(45^\circ) = \frac{H}{\text{Hypotenuse}} \]

Given that the hypotenuse (AB) is 6 units, we have:

\[ \sin(45^\circ) = \frac{H}{6} \]

Since \(\sin(45^\circ) = \frac{1}{\sqrt{2}}\), we can set up the equation:

\[ \frac{1}{\sqrt{2}} = \frac{H}{6} \]

### Step 3: Solve for the height (H).
To find H, we can rearrange the equation:

\[ H = 6 \cdot \frac{1}{\sqrt{2}} = \frac{6}{\sqrt{2}} \]

We can simplify this further by multiplying the numerator and denominator by \(\sqrt{2}\):

\[ H = \frac{6\sqrt{2}}{2} = 3\sqrt{2} \]

### Step 4: Identify the base of the parallelogram.
The base AD (or BC) is given as 8 units.

### Step 5: Calculate the area of the parallelogram.
The area (A) of a parallelogram is given by the formula:

\[ A = \text{Base} \times \text{Height} \]

Substituting the values we have:

\[ A = 8 \times 3\sqrt{2} = 24\sqrt{2} \]

### Conclusion:
The area of parallelogram ABCD is \(24\sqrt{2}\) square units.

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