Given:
Vertices of a quadrilateral LLGM are:
A(1,-2)
B(2,3)
C(k,2)
D(-4,-3)

To find:
The value of k

Approach:
To find the value of k, we can use the properties of a quadrilateral.

Properties of a quadrilateral:
1. The sum of the interior angles of a quadrilateral is equal to 360 degrees.
2. Opposite angles of a quadrilateral are equal.
3. The diagonals of a quadrilateral bisect each other.

Step 1: Find the coordinates of the diagonals.
The diagonals of the quadrilateral LLGM are AC and BD.

Coordinates of AC:
A(1,-2)
C(k,2)

Coordinates of BD:
B(2,3)
D(-4,-3)

Step 2: Find the midpoint of the diagonals.
The midpoint of the diagonals is the average of the coordinates of the endpoints.

Midpoint of AC:
(x1 + x2)/2, (y1 + y2)/2 = (1 + k)/2, (-2 + 2)/2 = (1 + k)/2, 0/2 = (1 + k)/2, 0

Midpoint of BD:
(x1 + x2)/2, (y1 + y2)/2 = (2 + (-4))/2, (3 + (-3))/2 = (-2)/2, 0/2 = -1, 0

Since the diagonals bisect each other, the midpoints of AC and BD should be the same.

(1 + k)/2 = -1
1 + k = -2
k = -3

Therefore, the value of k is -3.

Conclusion:
The value of k in the quadrilateral LLGM, with vertices A(1,-2), B(2,3), C(k,2), and D(-4,-3), is -3.