To demonstrate that triangles \( \triangle ABC \) and \( \triangle EFD \) are congruent (\( \triangle ABC \cong \triangle EFD \)) via rigid transformations, we can use a combination of translations, rotations, and reflections. Here’s a step-by-step outline of the sequence of transformations along with explanations for each step.
Step 1: Identify Corresponding Vertices
Before performing any transformations, we need to identify the corresponding vertices of the two triangles. Let’s assume the correspondence is as follows:
- \( A \) corresponds to \( E \)
- \( B \) corresponds to \( F \)
- \( C \) corresponds to \( D \)
This correspondence is typically based on the order of the vertices and may be confirmed by looking at the angles and side lengths of the triangles.
Step 2: Translation
- Translation: Move \( \triangle ABC \) such that point \( A \) lands on point \( E \).
- How: If the coordinates of \( A \) are \( (x_A, y_A) \) and the coordinates of \( E \) are \( (x_E, y_E) \), the translation vector would be \( (x_E - x_A, y_E - y_A) \).
- Result: After this translation, point \( A \) coincides perfectly with point \( E \).
Step 3: Rotate
- Rotation: Next, we need to rotate \( \triangle ABC \) around point \( E \) (now where \( A \) is located) so that point \( B \) aligns with point \( F \).
- How: Measure the angle \( \angle A \) and \( \angle E \). Determine the angle between line \( AE \) and line \( EF \). Rotate \( \triangle ABC \) by that angle.
- Result: After this rotation, point \( B \) will now overlap perfectly with point \( F \).
Step 4: Reflection (if necessary)
- Reflection (optional): If during the rotation, point \( C \) does not align with point \( D \) as expected, you might need to reflect \( \triangle ABC \) across the line that bisects angle \( E \) and \( F \).
- How: Identify the line of reflection based on the current alignment of \( A \) with \( E \), and \( B \) with \( F \).
- Result: After reflection, point \( C \) will coincide perfectly with point \( D \).
Conclusion
After completing these transformations—translation, rotation, and possibly reflection—we can conclude that each pair of corresponding vertices \( (A, E) \), \( (B, F) \), and \( (C, D) \) overlaps perfectly. Therefore, we have shown that \( \triangle ABC \cong \triangle EFD \).
Explanation of Overlapping
The congruence of triangles based on the corresponding vertices ensures that they are identical in shape and size. Through each transformation:
- The translation ensures that one vertex from each triangle matches.
- The rotation aligns a second vertex perfectly.
- The reflection (if necessary) guarantees that the third vertex coincides.
Thus, with rigid transformations, we maintain the properties of the triangles, leading to congruence.