To demonstrate that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent using a sequence of rigid transformations, we can follow these steps:
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Translation: Start by translating triangle \( \triangle ABC \) so that point \( A \) matches with point \( D \). A translation is a rigid transformation because it moves every point of the figure the same distance in the same direction, preserving the shape and size of the triangle.
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Rotation: After translating \( \triangle ABC \), rotate the triangle around point \( D \) until side \( AB \) aligns with side \( DE \). A rotation is another rigid transformation that maintains distances and angles, ensuring that the triangle remains congruent to its original position.
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Reflection (if necessary): If after the rotation, triangle \( ABC \) is still not aligned with triangle \( DEF \), you may need to reflect \( \triangle ABC \) across the line that bisects the angle at point \( D \) (or any line of reflection that aligns \( BC \) with \( EF \)). Reflection also preserves congruence.
Justification of Rigid Transformations:
Each of these transformations—translation, rotation, and reflection—are considered rigid transformations because they do not affect the size or shape of the triangle:
- Translation: Moves the triangle without altering its form.
- Rotation: Rotates around a fixed point without changing its side lengths or angles.
- Reflection: Flips the triangle across a line, preserving side lengths and angles.
ASA Congruence Theorem:
The sequence of transformations illustrates the ASA (Angle-Side-Angle) Congruence Theorem because:
- During the process, we maintain the congruence of two angles and the included side.
- For instance, if we know that \( \angle A \) and \( \angle B \) are congruent to \( \angle D \) and \( \angle E \), respectively, and that the included side \( AB \) is congruent to side \( DE \), then by ASA, the triangles must be congruent.
Thus, after performing these transformations, \( \triangle ABC \) will align perfectly with \( \triangle DEF \), confirming that \( \triangle ABC \cong \triangle DEF \) as stated by the ASA Congruence Theorem.
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