Question

Make a conjecture. How are rigid transformations and congruent figures related? Check all that apply.

Rigid transformations preserve segment lengths and angle measures.
Rigid transformations produce congruent figures.
If two figures are congruent, then there is a rigid transformation or a combination of rigid transformations that will map one onto the other.

Answer 1

All three statements are true — check them all.

1. Rigid transformations preserve segment lengths and angle measures.
- True. Rigid transformations (isometries: translations, rotations, reflections, and their compositions) preserve distances and angles by definition.

2. Rigid transformations produce congruent figures.
- True. Because they preserve distances and angles, the image of a figure under a rigid transformation has the same size and shape as the original, so it is congruent.

3. If two figures are congruent, then there is a rigid transformation or a combination of rigid transformations that will map one onto the other.
- True. In Euclidean geometry congruence is defined by existence of an isometry taking one figure to the other; for example, mapping three noncollinear vertices of one triangle to the corresponding three of another determines a rigid motion (possibly requiring a reflection if orientation is reversed).