Two trapezoids, upper A upper B upper C upper D and upper A prime upper B prime upper C prime upper D prime, are graphed on a coordinate plane.

To determine which statement is correct regarding the congruency of trapezoids \(ABCD\) and \(A'B'C'D'\), we need to consider the types of transformations involved:

1. **Reflection** (Statement #1) over the line \(y=x\): This transformation changes the coordinates of points \((x, y)\) to \((y, x)\). This can maintain congruency.
2. **Translation** (Statement #2): Moving a shape left or right, or up and down without rotation or reflection will maintain the shape and size, thus showing congruency.
3. **Rotation** (Statement #3): Rotating the trapezoid 270° counterclockwise (which is equivalent to 90° clockwise) would change the orientation but maintain congruency.
4. **Rotation** (Statement #4): A 90° counterclockwise rotation will also maintain the shape and size of the trapezoid.

Since translation does not change the orientation and only shifts the position, that would definitely confirm congruency. However, all of the transformations described can maintain congruency when applied correctly.

Without specific coordinates or a visual of the trapezoids, both Statement #2 (translation) and both rotation statements can potentially be correct under certain conditions.

If we must choose only one without additional context, **Statement #2** about the translation would be the most straightforward and universally applicable choice for congruency, as it does not involve any changes in orientation.

So, the answer is:

**Statement #2 is correct.**