To determine which explanation for the congruency of the trapezoids is correct, we need to evaluate the statements provided.
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Statement #1: The trapezoids are congruent because ABCD was reflected over the line \(y = x\).
- A reflection over the line \(y = x\) would switch the x and y coordinates of each vertex. If the trapezoids maintain their shape and size after this reflection, then this statement could be true.
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Statement #2: The trapezoids are congruent because ABCD was translated left 4 units and up 2 units.
- A translation moves all points of a figure the same distance in the same direction. Since translations do not change the shape or size of geometric figures, this statement is true if the translation was applied to trapezoid ABCD to create the other trapezoid.
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Statement #3: The trapezoids are congruent because ABCD was rotated 270° (counterclockwise).
- A rotation of 270° counterclockwise is equivalent to a 90° clockwise rotation. This would also maintain the side lengths and angles in the trapezoid, so this statement is plausible if one trapezoid can be obtained from the other via this rotation.
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Statement #4: The trapezoids are congruent because ABCD was rotated 90° (counterclockwise).
- A 90° counterclockwise rotation would also maintain the congruency. If this transformation applies to trapezoid ABCD and creates the other trapezoid while preserving dimensions and angles, this statement is true.
To conclude, any of the statements about transformations that preserve the congruency of the figures could be true, depending on the actual coordinates and the transformations applied.
To determine which statement is correct, you would need to examine the coordinates and the nature of the transformation that was applied. Without additional context or diagrams, we cannot definitively conclude which specific statement is correct. However, translations, rotations, and reflections all can ensure that the trapezoids are congruent if correctly applied.