How are rigid transformations used to justify the SAS Congruence Theorem?

Answer

Rigid transformations, such as translations, rotations, and reflections, justify the SAS (Side-Angle-Side) Congruence Theorem by showing that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the two triangles are congruent.

Explanation:

A rigid transformation is one that preserves distances and angles, meaning the shape and size of the figure are not altered. When you apply rigid transformations like translations (sliding), rotations (turning), and reflections (flipping) to one triangle, you can align it exactly with another triangle that has the same side lengths and the included angle. This shows that the two triangles are congruent, confirming the SAS Congruence Theorem. The theorem states that if two triangles have two corresponding sides and the included angle congruent, then the triangles are congruent. Rigid transformations help to visualize and prove this by ensuring that one triangle can be transformed to match the other without changing its size or shape.