Using the triangles shown, explain how the SAS congruence criteria follows from the definition of congruence in terms of rigid motions.
The SAS (side-angle-side) congruence criteria states that if two triangles have two sides and the included angle of one triangle congruent to two sides and the included angle of another triangle, then the triangles are congruent.
This can be explained using the definition of congruence in terms of rigid motions, which states that two figures are congruent if there exists a rigid motion (translation, rotation, or reflection) that maps one onto the other.
Consider two triangles, ΔABC and ΔDEF, where AB ≅ DE, AC ≅ DF, and ∠A ≅ ∠D. To show that they are congruent using rigid motions, we need to find a sequence of translations, rotations, and/or reflections that map one triangle onto the other.
First, we can translate triangle ΔABC so that point A coincides with point D. This translation preserves the lengths and angles of the sides, but changes the position of the triangle.
Next, we can rotate triangle ΔABC around point A by an angle that brings ∠A to coincide with ∠D. This rotation preserves the lengths and orientation of the sides, but changes the position of the triangle.
Finally, we can reflect triangle ΔABC across the line containing the segment BD. This reflection preserves the distances and angles between points, including the included angle ∠BAC or ∠EDF, but changes the orientation of the triangle.
After these rigid motions, triangle ΔABC is mapped onto triangle ΔDEF, and we can conclude that the two triangles are congruent by definition. Therefore, the SAS congruence criteria can be derived from the definition of congruence in terms of rigid motions.