Describe a sequence of rigid transformations that could be used to show that △ABC≅△DEF

Question

. 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.

Answer 1

To show that triangle \( \triangle ABC \) is congruent to triangle \( \triangle DEF \) using rigid transformations, we can follow a sequence of transformations: reflection, rotation, and/or translation. The crucial point is that each of these transformations preserves distances and angles, meaning they do not alter the shape or size of the triangles involved, thus maintaining the congruence.

Sequence of Rigid Transformations:

1. Reflection: Reflect triangle \( \triangle ABC \) across a line (for example, the line of symmetry between the two triangles if they are oriented in such a way). This reflection will produce a triangle \( \triangle A'B'C' \) that is congruent to triangle \( \triangle ABC \) because reflection is a rigid transformation.

2. Rotation: If necessary, rotate triangle \( \triangle A'B'C' \) around one of the vertices (say vertex \( A' \)) or the centroid, such that side \( A'B' \) aligns with side \( DE \). Since rotation preserves congruence, \( \triangle A'B'C' \) remains congruent to \( \triangle ABC \).

3. Translation: Finally, translate triangle \( \triangle A'B'C' \) so that vertex \( A' \) coincides with vertex \( D \), vertex \( B' \) with vertex \( E \), and so on. Translation is also a rigid transformation, so congruence is preserved.

How Each Transformation Preserves Congruence:

1. Reflection: This transformation creates a mirror image but retains the lengths of the sides and the measures of the angles, ensuring that \( \triangle A'B'C' \cong \triangle ABC \).

2. Rotation: Rotating the triangle around any point maintains its side lengths and angles, leading to \( \triangle A''B''C'' \cong \triangle A'B'C' \).

3. Translation: Moving the triangle across the plane to a new location does not affect side lengths or angles, so \( \triangle A''B''C'' \cong \triangle DEF \) after alignment.

ASA Congruence Theorem Explanation:

This sequence of rigid transformations illustrates the Angle-Side-Angle (ASA) Congruence Theorem. The ASA Theorem states that if two angles and the side between them in one triangle are congruent to two angles and the side between them in another triangle, then the triangles are congruent.

• During the reflection, it is guaranteed that the angles remain congruent (as reflected triangles maintain angle measures).
• When we rotate the triangle, we ensure that the corresponding angles remain aligned, maintaining the relationships dictated by the two angles and the side connecting them.
• The translation confirms the alignment of this angle-side-angle configuration.

Therefore, by reflecting, rotating, and translating triangle \( \triangle ABC \) such that it aligns exactly with triangle \( \triangle DEF \) while preserving angles and side lengths, we confirm that \( \triangle ABC \cong \triangle DEF \) as established by the ASA Congruence Theorem.