To determine which statement must be true after the rotation of parallelogram ABCD about the origin, let’s first analyze the properties of the rotation.
When we rotate a figure around the origin, the relative positions and orientation of points are preserved. Specifically, the angles and the parallelism of lines are maintained.
Given that line segments AB and CD are parallel, after a 45-degree counterclockwise rotation, the following statements can be evaluated:
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Line segment A'B' is parallel to line segment AB.
- After rotation, A'B' will be the image of AB, which means A'B' remains parallel to AB.
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Line segment A'D' is parallel to line segment B'C'.
- Since AD is parallel to BC and these segments will also rotate to their corresponding points, A'D' and B'C' will each be parallel to their original counterparts AD and BC, respectively. Hence, they will be parallel to each other as well.
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Line segment A'B' is parallel to line segment B'C'.
- A'B' represents one side of the parallelogram, and B'C' represents a different side. In fact, for a parallelogram, opposite sides are parallel, and adjacent sides are not. Therefore, this statement is not true.
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Line segment A'B' is parallel to line segment AD.
- A'B' represents one side of the parallelogram, and AD represents a different side. Thus, this statement is not true.
After evaluating these claims, the only statement that must be true following a 45-degree counterclockwise rotation of parallelogram ABCD is:
Line segment A'B' is parallel to line segment AB.
So the true statement is: A'B' is parallel to AB.
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