Use the image to answer the question.

Question

An illustration shows two triangles side by side. The two triangles are labeled A B C and D E F. Side A B is congruent to side E F. Side B C is congruent to side F D. Angle B is congruent to angle F.

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)

Answer 1

To show that triangles \( \triangle ABC \) and \( \triangle EFD \) are congruent using a sequence of rigid transformations, we can follow these steps:

1. Translation: First, translate triangle \( \triangle EFD \) so that vertex \( E \) coincides with vertex \( A \). This means we move triangle \( EFD \) horizontally or vertically until point \( E \) aligns with point \( A \).

2. Rotation: Next, rotate triangle \( \triangle EFD \) around point \( A \) (the new position of point \( E \)) so that angle \( F \) aligns with angle \( B \). Since angle \( B \) is congruent to angle \( F \), this rotation will ensure that point \( F \) now overlaps perfectly with point \( B \).

3. Final Positioning: After the rotation, point \( D \) will align with point \( C \) as well, because of the congruence of the sides \( BC \) and \( FD \).

Verification of Corresponding Vertices:

• The translation aligns \( E \) with \( A \).
• The rotation aligns \( F \) with \( B \).
• Given that side \( AB \) is congruent to side \( EF \) (by the given information), and after the rotation, side \( BC \) is congruent to side \( FD \), we can conclude that the points \( C \) and \( D \) must overlap perfectly based on the congruence of these sides. Hence, the vertices will correspond as follows:
  • \( A \) corresponds to \( E \)
  • \( B \) corresponds to \( F \)
  • \( C \) corresponds to \( D \)

SAS Congruence Theorem:

This sequence of transformations illustrates the Side-Angle-Side (SAS) Congruence Theorem because we have shown that:

• Two sides \( AB \) and \( EF \) are congruent (corresponding sides).
• The angle \( \angle B \) is congruent to angle \( \angle F\) (the included angle).
• The other pair of sides \( BC \) and \( FD \) are congruent (corresponding sides).

Since we have two pairs of corresponding sides that are congruent and the included angle is also congruent, we conclude that \( \triangle ABC \cong \triangle EFD \) by the SAS Congruence Theorem. Thus, the sequence of rigid transformations effectively demonstrates the congruence of the two triangles.