To analyze the properties of the parallelogram ABCD after it is rotated, we start by recalling the properties of a parallelogram: opposite sides are parallel.
Given that line segment AB is parallel to line segment CD, after a 45-degree counterclockwise rotation about the origin, we can determine the relationships between the line segments of the new figure A'B'C'D'.
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Statement A: Line segment A'B' is parallel to line segment AB.
- This statement is false because a rotation alters the direction of line segments. Therefore, A'B' will not be parallel to AB after the rotation.
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Statement B: Line segment A'D' is parallel to line segment B'C'.
- This statement is true, as after the rotation, A'D' (which is the new line segment after rotation of AD) will still be parallel to B'C' (the new line segment after rotation of BC) because they are still opposite sides of the parallelogram A'B'C'D' formed by the rotation.
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Statement C: Line segment A'B' is parallel to line segment B'C'.
- This statement is false since A'B' and B'C' are adjacent sides of the parallelogram, not opposite sides, and thus cannot be parallel.
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Statement D: Line segment A'B' is parallel to line segment AD.
- This statement is false because, after rotation, A'B' will not be parallel to the original line segment AD.
Based on this analysis, the statement that must be true is: B. Line segment A'D' is parallel to line segment B'C'.
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