Question

which sentence explains why polygon abcdefgh is congruent to polygon mnopqrst?
a. a sequence of translations will map mnopqrst onto abcdefgh.
b. a sequence of rigid transformations will map mnopqrst onto abcdefgh.
c. a sequence of rotations will map mnopqrst onto abcdefgh.
d. a sequence of reflections will map mnopqrst onto abcdefgh.

Explanation:

To determine why the polygons are congruent, we recall the definition of congruent polygons: they can be mapped onto each other using rigid transformations (translations, rotations, reflections), which preserve side lengths and angles.

• Option A: Translations alone may not be sufficient as the orientation might differ.
• Option B: Rigid transformations (translations, rotations, reflections) can map one polygon onto the other because rigid transformations preserve congruence (side lengths and angles remain the same).
• Option C: Rotations alone may not align the polygons correctly without other transformations.
• Option D: Reflections alone may not be enough, and the key is that any rigid transformation (not just reflections) can be used.

So the correct reasoning is that a sequence of rigid transformations will map one polygon onto the other.

Answer:

B. A sequence of rigid transformations will map MNOPQRST onto ABCDEFGH.