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..

Answer 1

To show that triangle \( \triangle ABC \cong \triangle DEF \) using a sequence of rigid transformations, we can use a combination of translations, rotations, and reflections. These transformations preserve the shape and size of the triangles, meaning that if the triangles are congruent, they will map onto each other exactly. Here’s how we can proceed:

Step 1: Translation

First, translate \( \triangle ABC \) so that one of its vertices (let’s say point A) coincides with the corresponding vertex in \( \triangle DEF \) (point D). This is a rigid transformation, which means that it keeps all angle measures and side lengths unchanged.

Step 2: Rotation

Next, we can rotate \( \triangle ABC \) about point D (which is now the location of point A after the translation) so that another vertex of \( \triangle ABC \) (let’s say point B) aligns with the corresponding vertex of \( \triangle DEF \) (point E). Rotations also maintain the size and shape of the triangle, meaning all angles and sides remain congruent.

Step 3: Reflection (if necessary)

If after the rotation point C does not coincide with point F, we can perform a reflection over the line that runs through points D and E (or any appropriate line that ensures that C aligns with F). A reflection is another type of rigid transformation that preserves distances and angles.

Final Confirmation

After applying these transformations, we map \( \triangle ABC \) onto \( \triangle DEF \) such that:

• Point A maps to point D
• Point B maps to point E
• Point C (after proper reflection, if needed) maps to point F

How These Transformations Ensure Congruence

Each transformation (translation, rotation, and reflection) preserves the distances between points and the measures of the angles. Therefore, \( \triangle ABC \) is congruent to \( \triangle DEF \) through these transformations.

ASA Congruence Theorem Illustration

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 case, as we assume that angles A, B, and the included side (AB or AC) are congruent to angles D, E, and the included side (DE or DF) respectively, the transformations illustrate how we can align these two triangles to demonstrate their congruence through rigid motions.

Thus, the sequence of transformations (translation, rotation, and potentially reflection) directly showcases that triangle \( \triangle ABC \) is congruent to triangle \( \triangle DEF \) and illustrates the principles behind the ASA Congruence Theorem effectively.