In order to prove that triangle ABC is congruent to triangle EFD, we must use a series of rigid transformations. Rigid transformations, including translations, rotations, and reflections, preserve the shape and size of an object.
First, we analyze the given triangles, ABC and EFD, and identify the corresponding vertices that will perfectly overlap to demonstrate congruence:
Triangle ABC:
- Vertex A corresponds to vertex E.
- Vertex B corresponds to vertex F.
- Vertex C corresponds to vertex D.
To prove that each pair of corresponding vertices will overlap perfectly, we need to explain how each rigid transformation will maintain congruence:
1. Translation: By translating triangle ABC to the right or left, we can make vertex A coincide perfectly with vertex E while preserving distance and angle measures.
2. Rotation: Rotating triangle ABC around a point allows the corresponding vertices to overlap perfectly, preserving distance and angle measures.
3. Reflection: Reflecting triangle ABC over a given line results in corresponding vertices overlapping perfectly while maintaining distances and angle measures.
The sequence of rigid transformations may involve a combination of translations, rotations, and reflections to show the congruence of triangle ABC and triangle EFD.
Applying the SAS Congruence Theorem, if two triangles have two pairs of corresponding angles that are congruent and their included side lengths are also congruent, the triangles are congruent. Once we have demonstrated that the three pairs of corresponding vertices overlap perfectly, we can apply the SAS congruence condition. The corresponding angles and included side lengths of triangle ABC and triangle EFD will be congruent due to the rigid transformations. Therefore, triangle ABC is congruent to triangle EFD.
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To show that △ABC≅△EFD, we need to demonstrate that the two triangles are congruent by using a sequence of rigid transformations. Rigid transformations include translations, rotations, and reflections, which preserve the shape and size of an object.
First, let's examine the given triangles △ABC and △EFD. We need to identify the corresponding vertices that will overlap perfectly to prove congruence.
△ABC:
- Vertex A corresponds to vertex E.
- Vertex B corresponds to vertex F.
- Vertex C corresponds to vertex D.
To prove that each pair of corresponding vertices will overlap perfectly, we need to explain how each transformation will preserve congruence.
1. Translation:
By translating triangle △ABC to the right or left, we can make vertex A coincide perfectly with vertex E. Translation preserves distance and angle measures, ensuring that the two vertices will overlap exactly.
2. Rotation:
If we rotate triangle △ABC around a point, the corresponding vertices can overlap perfectly. A rotation preserves both distance and angle measures, guaranteeing that the two triangles will have congruent angles and side lengths.
3. Reflection:
Reflecting triangle △ABC over a given line will result in corresponding vertices overlapping perfectly. Reflection preserves distances and angle measures, ensuring congruent triangles.
The sequence of rigid transformations that can be used to show △ABC≅△EFD may involve a combination of translations, rotations, and reflections.
By using the SAS Congruence Theorem, we can conclude that if two triangles have two pairs of corresponding angles that are congruent and their included side lengths are also congruent, the triangles are congruent. In this case, once we have demonstrated that the three pairs of corresponding vertices overlap perfectly, we can apply the SAS congruence condition. The corresponding angles and included side lengths of △ABC and △EFD will be congruent due to the rigid transformations applied. Therefore, △ABC≅△EFD.
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