To prove that the quadrilateral ABCD is a parallelogram, we need to show that the opposite sides are parallel. This can be done by proving that the slopes of opposite sides are equal.

Step 1: Find the slope of AB and CD.

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Slope of AB:

The coordinates of A and B are $(-2, 3)$ and $(6, 1)$, respectively.

$\text{slope of AB} = \frac{1 - 3}{6 - (-2)} = \frac{-2}{6 + 2} = \frac{-2}{8} = -\frac{1}{4}$

Slope of CD:

The coordinates of C and D are $(5, -2)$ and $(-3, 0)$, respectively.

$\text{slope of CD} = \frac{0 - (-2)}{-3 - 5} = \frac{2}{-8} = -\frac{1}{4}$

Since the slopes of AB and CD are equal $(-1/4)$, the sides AB and CD are parallel.

Step 2: Find the slope of AD and BC.

Slope of AD:

The coordinates of A and D are $(-2, 3)$ and $(-3, 0)$, respectively.

$\text{slope of AD} = \frac{0 - 3}{-3 - (-2)} = \frac{-3}{-3 + 2} = \frac{-3}{-1} = 3$

Slope of BC:

The coordinates of B and C are $(6, 1)$ and $(5, -2)$, respectively.

$\text{slope of BC} = \frac{-2 - 1}{5 - 6} = \frac{-3}{-1} = 3$

Since the slopes of AD and BC are equal (3), the sides AD and BC are parallel.

Conclusion:

Since opposite sides AB and CD, as well as AD and BC, are parallel, quadrilateral ABCD is a parallelogram.

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Would you like more details or have any questions? Here are some related questions to explore:

1. How do you determine if a quadrilateral is a rectangle?
2. How do you calculate the area of a parallelogram given four points?
3. What are the properties of a parallelogram?
4. How do you find the lengths of the sides of a parallelogram using coordinates?
5. Can a parallelogram be inscribed in a circle?

Tip: When checking for parallel lines in coordinate geometry, always compare the slopes of opposite sides.