Describe a sequence of rigid transformations that could be used to show that △ABC≅△EFD . Be sure to explain how you know that each pair of corresponding vertices will overlap perfectly. Then, explain why this sequence illustrates the SAS Congruence Theorem. (4 points)
To show that triangles △ABC and △EFD are congruent, we need to perform a sequence of rigid transformations.
1. Translation: We can start by translating triangle △EFD so that its vertex matches up with the vertex of triangle △ABC. We can shift triangle △EFD horizontally or vertically until point E overlaps with point A.
2. Rotation: Once point E coincides with point A, we can rotate triangle △EFD around the common vertex, such as point A, until either point F overlaps with point B or point D overlaps with point C.
3. Translation: Finally, we can translate the rotated and overlapped triangle △EFD horizontally or vertically until the third vertex, either D or F, coincides with the corresponding vertex in triangle △ABC.
We know that each pair of corresponding vertices will overlap perfectly because we start by aligning one vertex of triangle △EFD with the corresponding vertex of triangle △ABC. This ensures that E and A overlap exactly. Then, by rotating the triangle around the common vertex, we align either F or D with the corresponding vertex in triangle △ABC. The rotation around a fixed point preserves the shape and size of the triangle, ensuring that the other two vertices, B and C, will coincide with their corresponding vertices in triangle △ABC.
This sequence of rigid transformations illustrates the SAS (Side-Angle-Side) Congruence Theorem because we have shown that the corresponding sides and the included angle of the two triangles are congruent. In the first step, the translation ensures that side EA is congruent to side AB. In the second step, the rotation ensures that the included angle at vertex A in triangle △ABC is congruent to the included angle at the common vertex in triangle △EFD. Lastly, in the third step, the translation ensures that either side FD or side DE is congruent to either side BC or side CB, respectively. Since we have proven that the corresponding sides and included angles are congruent, we can conclude that triangles △ABC and △EFD are congruent by the SAS Congruence Theorem.