To show that triangles △ABC and △EFD are congruent using a sequence of rigid transformations, we can follow these steps:
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Translation: First, we can translate triangle △EFD so that point E coincides with point A. This movement shifts the position of triangle EFD without altering its shape or size.
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Rotation: Next, we will rotate triangle △EFD around point A (which is now the location of point E in triangle EFD) such that side EF (now located at point A) aligns with side AB of triangle △ABC. This rotation can be performed in such a way that angle F (which is congruent to angle B) also matches angle B at its location.
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Final Adjustment: After the rotation, we could check the alignment of side FD with side BC, ensuring that they also overlap perfectly due to the earlier congruence statements of the sides and angles.
Correspondence of Vertices: After performing these transformations, the corresponding vertices will overlap as follows:
- Point A (of △ABC) will correspond to point E (of △EFD).
- Point B (of △ABC) will correspond to point F (of △EFD).
- Point C (of △ABC) will correspond to point D (of △EFD).
Justification of Overlapping: Each pair of corresponding vertices overlaps perfectly because:
- The translation ensures that corresponding positions move to align correctly.
- The rotation matches the orientation of the triangles while preserving their shape, hence maintaining relationships between angles and sides.
Illustration of the SAS Congruence Theorem: This sequence of transformations illustrates the SAS (Side-Angle-Side) Congruence Theorem. According to this theorem, two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle. In this scenario:
- Side AB is congruent to side EF.
- Side BC is congruent to side FD.
- Angle B is congruent to angle F.
Thus, by demonstrating that we can perfectly align the triangles through translation and rotation while preserving the relationships of sides and angles, we have shown that △ABC is congruent to △EFD, satisfying the conditions of the SAS Congruence Theorem.
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