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..

To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle EFD \) using rigid transformations, we can employ a sequence of translations, rotations, and reflections. Let's outline a possible sequence of transformations and explain how corresponding vertices will overlap perfectly.

1. Translation: Begin by translating triangle \( \triangle ABC \) so that point \( A \) coincides with point \( E \). This means that the entire triangle moves along a straight path without changing its shape or orientation. After this transformation, point \( A \) is now at point \( E \).

2. Rotation: Next, apply a rotation about point \( E \) (now the position of point \( A \)) so that point \( B \) aligns with point \( F \). The angle of rotation will depend on the initial positions of triangle \( ABC \) and point \( F \).

3. Reflection (if necessary): If after the previous rotation, point \( C \) does not align with point \( D \), then a reflection across a line can be applied. The line of reflection could be the line that bisects the angle formed by line segments \( EF \) and the segment now formed by \( EB \).

After applying these transformations, we guarantee that each pair of corresponding vertices will overlap perfectly. This is because:

• The translation ensured that point \( A \) aligned with point \( E \).
• The rotation adjusted the orientation of the triangle so that point \( B \) now matches point \( F \).
• The reflection (if applied) would adjust point \( C \) to line up perfectly with point \( D \).

Correspondence of Vertices

After these transformations:

• \( A \) coincides with \( E \)
• \( B \) coincides with \( F \)
• \( C \) coincides with \( D \)

This means that all corresponding vertices are aligned, and thus \( \triangle ABC \) is congruent to \( \triangle EFD \).

SAS Congruence Theorem

This sequence illustrates the SAS (Side-Angle-Side) Congruence Theorem, which states that if two triangles have two sides and the included angle of one triangle equal to the corresponding two sides and included angle of the other triangle, then the triangles are congruent.

In our case:

• The sides \( AB \) and \( AC \) correspond to sides \( EF \) and \( ED \).
• The angle \( \angle ABC \) corresponds to angle \( \angle EFD \).
• The congruence of the sides and angles has been preserved through rigid transformations.

Thus, by demonstrating that corresponding sides and angles are equal using these transformations, we have shown that \( \triangle ABC \) is congruent to \( \triangle EFD \) using the SAS Congruence Theorem.