Use the image to answer the 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 △ABC ≅ △EFD using rigid transformations and the SAS Congruence Theorem (Side-Angle-Side), we can follow these steps:

1. Identify Corresponding Parts:

   • From the given information, we know that:
     • Side \( AB \) is congruent to side \( EF \).
     • Side \( BC \) is congruent to side \( FD \).
     • Angle \( \angle B \) is congruent to angle \( \angle F \).

   Therefore, we can pair the vertices as follows:

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

2. Sequence of Rigid Transformations:

   • Step 1: Translation: First, translate △ABC so that point \( B \) aligns with point \( F \). This aligns angle \( \angle B \) with angle \( \angle F \) perfectly since they are congruent.
   • Step 2: Rotation: Once both triangles share the same vertex \( B \) (at \( F \)), we can rotate triangle ABC around point \( B \) (now positioned at point \( F \)) until side \( AB \) is aligned with side \( EF \). Since \( AB \) is congruent to \( EF \), they will overlap perfectly.
   • Step 3: Final Position: After aligning \( AB \) with \( EF \), the side \( BC \) will automatically align with side \( FD \) because of the congruence of the sides.

3. Justification of Corresponding Vertices:

   • After applying the translation and rotation, point \( A \) will now occupy the same position as point \( E \), point \( B \) will occupy the same position as point \( F \), and point \( C \) will occupy the same position as point \( D \). Therefore, all corresponding vertices will overlap perfectly.

4. Illustration of SAS Congruence Theorem:

   • The transformation sequence illustrates the SAS Congruence Theorem, which states that if two sides and the angle between them in one triangle are congruent to two sides and the angle between them in another triangle, then the triangles are congruent.
   • In our case:
     • Side \( AB \) is congruent to side \( EF \) (first pair of sides).
     • Angle \( \angle B \) is congruent to angle \( \angle F \) (the angle included between the two pairs of sides).
     • Side \( BC \) is congruent to side \( FD \) (second pair of sides).
   • Thus, △ABC is congruent to △EFD by SAS, as we have shown that the two sides surrounding the angle and the angle itself are congruent, confirming triangle congruence.

By following this sequence of transformations and verifying the conditions of the SAS Congruence Theorem, we can conclusively show that △ABC ≅ △EFD.