To demonstrate that △ABC is congruent to △EFD using rigid transformations, we can apply a sequence of transformations: translation, rotation, and potentially reflection. Here’s a step-by-step explanation of how to perform these transformations and why the corresponding vertices will overlap perfectly.
Step 1: Identify Corresponding Vertices
Assume we have the vertices of the triangles as follows:
- A corresponds to E
- B corresponds to F
- C corresponds to D
Step 2: Translation
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Translation of Triangle ABC: Move triangle ABC so that vertex A coincides with vertex E. This can be achieved by translating every point in triangle ABC such that A lands on E.
- Overlap: After the translation, point A aligns with point E (A→E).
Step 3: Rotation
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Rotation of Triangle ABC: Now that A is at E, we need to position the triangle so that the side AB corresponds to EF. To do this, rotate triangle ABC around point E (now coinciding with A) until side AB aligns with side EF. The angle of rotation is determined by the angle between these two segments.
- Overlap: After the rotation, point B will now line up perfectly with point F (B→F).
Step 4: Verify Vertex C
- Check Vertex C: At this point, vertex C will be positioned according to the rotation. By the properties of rigid transformations and given that the triangle's sides and angles remain unchanged, side AC will now overlap with side ED, and vertex C will coincide with vertex D (C→D).
Conclusion on Vertex Correspondence:
After completing the translation and rotation:
- Vertex A aligns with E (A→E)
- Vertex B aligns with F (B→F)
- Vertex C aligns with D (C→D)
Thus, all pairs of corresponding vertices overlap perfectly, demonstrating that △ABC can be transformed into △EFD through these rigid transformations.
Explanation of SAS Congruence Theorem:
The SAS (Side-Angle-Side) Congruence Theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
In our transformations:
- SAS Conditions:
- We translated triangle ABC to have side AB congruent and positioned correctly with side EF.
- The included angle ∠ABC is congruent to angle ∠EFD since rotation preserves angle measures.
- Finally, side AC is congruent to side ED while being positioned as per the rotation.
Through this sequence of transformations, we've shown that all corresponding parts of triangles ABC and EFD are congruent, thereby illustrating the SAS Congruence Theorem effectively.