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. rotation and translation

Answer 1

To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle EFD \) using rigid transformations, we can utilize a sequence of a rotation followed by a translation. Here’s how we can accomplish this:

Step 1: Identify Corresponding Parts

First, we need to identify which vertices and sides correspond between the two triangles. For example, we can assume:

• Vertex \( A \) corresponds to vertex \( E \)
• Vertex \( B \) corresponds to vertex \( F \)
• Vertex \( C \) corresponds to vertex \( D \)

Step 2: Check Side Lengths

Next, we should ensure that the side lengths of the triangles satisfy the Side-Angle-Side (SAS) condition. This means at least one pair of sides in \( \triangle ABC \) has the same length as the corresponding side in \( \triangle EFD \), and the angle included between these sides is also congruent. Specifically, we need to check:

• \( AB \) corresponds to \( EF \)
• \( AC \) corresponds to \( ED \)
• The angle \( \angle ABC \) corresponds to \( \angle EFD \)

Step 3: Performing the Rigid Transformations

1. Rotation:

   • Rotate triangle \( \triangle ABC \) around point \( A \) (or \( E \) if we align it that way on the coordinate plane) by the angle necessary to align side \( AB \) with \( EF \). This action will ensure that point \( B \) lines up directly with point \( F \) because the side lengths and the angles remain the same due to the rigid nature of rotations.

2. Translation:

   • After the rotation, we will translate triangle \( \triangle ABC \) such that point \( A \) aligns with point \( E \). This translation will also move point \( B \) to point \( F \) and point \( C \) to point \( D \) because they have already been positioned correctly through the rotation.

Overlapping of Corresponding Vertices

• After completing the rotation, \( B \) will overlap \( F \), and \( C \) will overlap \( D \) when we translate \( A \) to \( E \). Thus, all pairs of corresponding vertices \( (A, E), (B, F), (C, D) \) will coincide perfectly, demonstrating that \( \triangle ABC \) and \( \triangle EFD \) occupy the same space geometrically.

SAS Congruence Theorem

This sequence of transformations illustrates the SAS Congruence Theorem, which states that if two sides of one triangle are congruent to two sides of another triangle, and the included angles are congruent, then the triangles are congruent.

• In our case, we have:
  • Side \( AB \) congruent to side \( EF \)
  • Side \( AC \) congruent to side \( ED \)
  • The included angle \( \angle ABC \) congruent to \( \angle EFD \)

Since the sequence of transformations preserves distance and angles, we can conclude that the triangles \( \triangle ABC \) and \( \triangle EFD \) are congruent, fulfilling the requirements of the SAS Congruence Theorem.