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. reflection

Answer 1

To demonstrate that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent, we can use a sequence of rigid transformations, such as reflections, rotations, and translations. Rigid transformations preserve the shape and size of figures, meaning the resulting triangle will be congruent to the original. Here's how you could show the congruence using these transformations, specifically focusing on reflection as part of the sequence:

Sequence of Rigid Transformations:

1. Reflection: Start by reflecting triangle \( \triangle ABC \) across the line that bisects angle \( A \) (let's assume it reflects to a position where point \( A \) aligns with point \( D \)). This transformation keeps the lengths of the sides and angles the same, ensuring that the shape remains congruent.

2. Rotation: After the reflection, rotate triangle \( \triangle ABC \) (now reflected to a new triangle \( \triangle A'B'C' \)) around point \( A' \) (which is now aligned with \( D \)) to align point \( B' \) with point \( E \). This rotation maintains congruence since we are turning the figure around a point without altering its size or shape.

3. Translation: Finally, translate triangle \( \triangle A'B'C' \) so that point \( C' \) coincides with point \( F \). Again, this transformation does not change the shape or size of the triangle, preserving congruence.

Reasoning for Each Transformation:

• Reflection: The reflection across the angle bisector preserves distance and angle measures, ensuring the result is congruent to the original triangle.

• Rotation: Rotating at point \( A' \), here acting as the new vertex, maintains side lengths and angles, thus keeping \( \triangle A'B'C' \) congruent to \( \triangle ABC \).

• Translation: Translational movements carry all points the same distance in a given direction, ensuring congruence remains intact.

Illustrating ASA Congruence Theorem:

The sequence of transformations illustrates the ASA (Angle-Side-Angle) Congruence Theorem, which 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 two triangles are congruent.

1. Reflection establishes that angle \( A \) in \( \triangle ABC \) matches angle \( D \) in \( \triangle DEF \), confirming one pair of angles is congruent.

2. Rotation and the configuration ensure that the included side \( AC \) (which became \( A'C' \) in the new reflected triangle) is congruent to \( DE \).

3. Translation solidifies that the corresponding vertices match up perfectly, maintaining the congruency of the remaining angles.

By completing this sequence, \( \triangle ABC \) has been transformed to exactly overlap with \( \triangle DEF \), demonstrating their congruence through rigid transformations and affirming the ASA Congruence Theorem.