To determine the angle of rotation for the quadrilateral $A'B'C'D'$, which is the image of quadrilateral $ABCD$ under a rotation about the origin $(0, 0)$, we can use the following process:

Steps:

1. Analyze the coordinates: We need to identify whether any of the original points $A, B, C, D$ and their corresponding image points $A', B', C', D'$ are given or can be estimated from the grid. However, if they are not directly provided, we can reason based on the standard properties of rotations.

2. Rotation properties:

   • A 90° rotation counterclockwise (positive direction) swaps the coordinates of a point and changes the sign of the new $x$-coordinate. Thus, $(x, y) \to (-y, x)$.
   • A -90° rotation (clockwise) also swaps the coordinates, but changes the sign of the new $y$-coordinate: $(x, y) \to (y, -x)$.
   • A 30° or -30° rotation is more complex and involves trigonometric transformations, which are less likely for a grid-based problem unless detailed coordinates are given.

3. Matching the rotation type: Without specific coordinates, the problem typically suggests a grid-based rotation, meaning the answer is likely to involve a common rotation angle such as $90^\circ$ or $-90^\circ$.

Conclusion:

Given the options and the typical behavior of common rotations, the answer is likely Choice A: $-90^\circ$, which corresponds to a clockwise rotation.

Would you like further clarification or details on the rotation process?

Here are some follow-up questions:

1. How does a positive vs. negative rotation affect the direction of rotation?
2. What are the properties of a 180° rotation?
3. How can trigonometric functions describe general rotations in the plane?
4. What happens to the orientation of a figure after a 90° rotation?
5. How can you determine the center of rotation if it's not the origin?

Tip: Always check whether rotation preserves distances and angles, since those are key properties of rigid transformations like rotations.