To demonstrate that triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent using a series of rigid transformations and the ASA (Angle-Side-Angle) Congruence Theorem, we start by establishing the correspondence between their parts, with \( \angle A \) matching \( \angle D \), \( \angle B \) matching \( \angle E \), and the side \( \overline{AB} \) corresponding to \( \overline{DE} \). Initially, we perform a translation to move triangle \( \triangle ABC \) such that point \( A \) aligns with point \( D \) and point \( B \) coincides with point \( E \). Following this, we apply a rotation around point \( D \) to ensure that \( \angle A \) perfectly aligns with \( \angle D \), which concurrently positions point \( C \) at point \( F \) due to the alignment of the corresponding angles. Throughout these transformations, we maintain the congruence of the triangles, as translations and rotations do not alter their shape or size. Finally, with both angles \( \angle A \) and \( \angle B \) congruent to \( \angle D \) and \( \angle E \) respectively, and \( \overline{AB} \) congruent to \( \overline{DE} \) as the included side, we affirm that \( \triangle ABC \cong \triangle DEF \) in accordance with the ASA theorem, demonstrating that two angles and the included side are congruent, thereby establishing their overall congruence.
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To demonstrate that \( \triangle ABC \cong \triangle DEF \) using a sequence of rigid transformations and to illustrate the ASA (Angle-Side-Angle) Congruence Theorem, you can follow these steps:
### Step 1: Identify Corresponding Parts
Before initiating any transformations, determine how the triangles correspond:
- Assume \( \angle A \) corresponds to \( \angle D \)
- Assume \( \angle B \) corresponds to \( \angle E \)
- The included side \( \overline{AB} \) corresponds to \( \overline{DE} \)
### Step 2: Rigid Transformations
The sequence of rigid transformations that will show \( \triangle ABC \cong \triangle DEF \) could involve the following:
1. **Translation**:
- Translate \( \triangle ABC \) such that point \( A \) coincides with point \( D \) and point \( B \) coincides with point \( E \). This aligns two of the corresponding vertices perfectly, placing \( A \) on \( D \) and \( B \) on \( E \).
2. **Rotation**:
- Rotate \( \triangle ABC \) around point \( D \) such that \( \angle A \) aligns perfectly with \( \angle D \). This rotation will allow us to position all corresponding angles to overlap.
- As a result of this transformation, point \( C \) will move into the position of point \( F \), given that \( \angle B \) aligns with \( \angle E \) after the transformation.
### Step 3: Ensure the Result is Congruent
- **Translation**:
- Translations do not alter the shape or size of the triangle; they merely shift it in space. As such, the resulting figure remains congruent to \( \triangle ABC \).
- **Rotation**:
- Rotations also do not change the size or shape of a figure; they merely rotate it around a specific point. Hence, after the rotation, the angles and sides of \( \triangle ABC \) remain congruent to the corresponding parts of \( \triangle DEF \).
### Step 4: Illustrating the ASA Congruence Theorem
- **Angle-Side-Angle**:
- After completing the above transformations, you will have:
- \( \angle A \) (corresponding to \( \angle D \)) is congruent,
- \( \angle B \) (corresponding to \( \angle E \)) is congruent, and
- The side \( \overline{AB} \) (corresponding to \( \overline{DE} \)) which is between the angles, is congruent.
Thus, we have established two angles and the included side as congruent: this fits the criteria for the ASA theorem.
### Conclusion
Through a sequence of rigid transformations involving translation followed by rotation, the corresponding vertices of \( \triangle ABC \) and \( \triangle DEF \) can be made to overlap perfectly. This validates that \( \triangle ABC \cong \triangle DEF \) and illustrates the ASA Congruence Theorem since we have demonstrated that two angles and the included side are congruent in both triangles.
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