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Let's begin by analyzing the given quadrilateral \( ABCD \) and addressing the two questions step by step.

### Step 1: Verify if the quadrilateral is a parallelogram (for question 1).

The coordinates of the vertices are:
- \( A = (3, 4) \)
- \( B = (4, -4) \)
- \( C = (-3, -8) \)
- \( D = (-4, 0) \)

A quadrilateral is a parallelogram if the opposite sides are both parallel. We can check this by calculating the slopes of opposite sides:

1. **Slope of \( AB \)**:
\[
\text{slope of } AB = \frac{-4 - 4}{4 - 3} = \frac{-8}{1} = -8
\]

2. **Slope of \( CD \)**:
\[
\text{slope of } CD = \frac{0 - (-8)}{-4 - (-3)} = \frac{8}{-1} = -8
\]
Thus, \( AB \parallel CD \).

3. **Slope of \( BC \)**:
\[
\text{slope of } BC = \frac{-8 - (-4)}{-3 - 4} = \frac{-4}{-7} = \frac{4}{7}
\]

4. **Slope of \( AD \)**:
\[
\text{slope of } AD = \frac{0 - 4}{-4 - 3} = \frac{-4}{-7} = \frac{4}{7}
\]
Thus, \( BC \parallel AD \).

Since opposite sides are parallel, quadrilateral \( ABCD \) is a parallelogram.

**Conclusion for question 1:**  
The quadrilateral is a parallelogram. The first statement is **true**.

---

### Step 2: Verify the area of the parallelogram (for question 2).

The area of a parallelogram can be calculated using the formula:

\[
\text{Area} = \left| \vec{AB} \times \vec{AD} \right|
\]

We first calculate the vectors \( \vec{AB} \) and \( \vec{AD} \):
- \( \vec{AB} = (4 - 3, -4 - 4) = (1, -8) \)
- \( \vec{AD} = (-4 - 3, 0 - 4) = (-7, -4) \)

Now, calculate the cross product \( \vec{AB} \times \vec{AD} \):
\[
\vec{AB} \times \vec{AD} = (1)(-4) - (-8)(-7) = -4 - 56 = -60
\]
The area is the absolute value of the cross product:
\[
\text{Area} = | -60 | = 60
\]

The second statement claims that the area is \( 30\sqrt{5} \). To verify if these are equivalent, we compare:
\[
30\sqrt{5} \approx 67.08
\]
Since \( 60 \neq 30\sqrt{5} \), the second statement is **false**.

---

### Final Answers:
1. The quadrilateral is a parallelogram: **True**.
2. The area is \( 30\sqrt{5} \): **False**.

Would you like more details or have any further questions?

Here are some related questions:
1. How do you prove the diagonals bisect each other in a parallelogram?
2. How do you calculate the perimeter of a parallelogram given its vertices?
3. What are the properties of a rhombus, and how does it relate to a parallelogram?
4. How can you calculate the area of any convex quadrilateral using its vertices?
5. Can the same method for calculating area be used for non-parallelograms?

**Tip:** Always verify slopes or distances between points to confirm geometric properties like parallelism or equal lengths.

This problem involves verifying if a given quadrilateral is a parallelogram and calculating its area. Using coordinate geometry, we determine parallel sides and apply the vector cross product to find the area. Suitable for students in Grades 9-12.

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