The first image represents the parallelogram $ABCD$ being rotated $270^\circ$ counterclockwise about the origin, resulting in $A'B'C'D'$. The second image provides statements to verify if they are true about the transformed parallelogram.

Analysis of the Statements:

1. $A'B'$ is parallel to $D'C'$:

   • True. Parallelism is preserved under rotation since it is a rigid motion.

2. $A'D'$ is the same length as $AD$:

   • True. Distances between points remain unchanged under rotation.

3. $\angle A$ has the same measure as $\angle A'$:

   • True. Angles within a figure remain unchanged under rigid transformations.

4. $A'$ has the same coordinates as $A$:

   • False. The coordinates of $A$ change after a rotation of $270^\circ$.

Conclusion:

The first three statements are true, and the last one is false.

Would you like to explore further explanations or coordinate transformations in detail? Let me know!

Relative Questions:

1. What are the coordinates of $A'$, $B'$, $C'$, and $D'$ after the rotation?
2. How is a $270^\circ$ counterclockwise rotation mathematically defined in terms of coordinates?
3. How does rigid motion affect the properties of geometric shapes?
4. Can other transformations (e.g., reflection or translation) also preserve angle measures and side lengths?
5. How would a $90^\circ$ clockwise rotation differ from a $270^\circ$ counterclockwise rotation in terms of final coordinates?

Tip:

When performing rotations, always use the rotation formula:
$(x, y) \rightarrow (-y, x)$ for $270^\circ$ counterclockwise.