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Question

Two triangles are labeled upper A upper B upper C, and upper F upper E upper D.

Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF . Explain how you know each transformation will result in a congruent figure. Then, explain how this sequence of rigid transformations illustrates the ASA Congruence Theorem.

(4 points)

Answer 1

To demonstrate that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent (\( \triangle ABC \cong \triangle DEF \)), we can use a sequence of rigid transformations. Rigid transformations include translations, rotations, and reflections, which maintain the size and shape of the figures involved. Here's how you could proceed:

Sequence of Rigid Transformations

1. Translation: Start by translating triangle \( DEF \) so that point \( D \) moves to the position of point \( A \). This shifts the entire triangle without changing its shape or size, preserving congruence.

2. Rotation: After translation, rotate triangle \( DEF \) about point \( A \) until side \( DE \) aligns with side \( AB \). This transformation also keeps the distances between points the same, ensuring congruence is maintained.

3. Reflection (if necessary): If the orientation is unaligned after the rotation, you can reflect triangle \( DEF \) over line \( AB \) or any appropriate axis. This step will not change the size or angles and will help in correctly positioning triangle \( DEF \) over \( \triangle ABC \).

Explanation of Congruence

Each of these transformations is a rigid transformation, meaning they do not distort the shapes involved:

• Translation keeps all distances and angles unchanged.
• Rotation affects the orientation but keeps distances between points the same and preserves the angles.
• Reflection flips the shape over a given line, maintaining congruence.

Illustration of the ASA Congruence Theorem

The ASA (Angle-Side-Angle) Congruence Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

• In this setup, once \( \triangle DEF \) has been translated, rotated, and possibly reflected, you can identify two pairs of angles and the included side:
  • For example, if \( \angle A \) and \( \angle B \) in triangle \( ABC \) correspond to \( \angle D \) and \( \angle E \) in triangle \( DEF \), and if \( AB \) corresponds to \( DE \) as the included side, then according to ASA, we can conclude that \( \triangle ABC \cong \triangle DEF \).

Conclusion

Through the outlined sequence of rigid transformations and the application of the ASA Congruence Theorem, we can confirm that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent. Each transformation maintains congruence, allowing us to rotate, translate, or reflect one triangle to match another without altering the angles and lengths, reinforcing the relationship stated in the theorem.