Final answer:
The side lengths of triangles A'B'C' and ABC remain equal during a rotation because rotations are rigid transformations that preserve distances and angles within a shape.
Step-by-step explanation:
When a triangle A'B'C' coincides with another triangle ABC during a rotation, the measures of the side lengths of both triangles remain unchanged. This demonstrates a fundamental property of rotations in geometry: they are rigid transformations. A rigid transformation preserves the lengths of sides and angles of a shape. Therefore, regardless of the degree or direction of rotation, as long as triangle A'B'C' coincides with triangle ABC, the side lengths and angles will be the same.
Rotations are considered isometric transformations, meaning that all distances between corresponding points in the original and the image are equal post-transformation; rotation neither distorts the shape nor alters its size. This property is pivotal in understanding rotational symmetry in various geometric figures such as cubes, which have multiple rotational axes that leave its appearance unchanged.