Describe a sequence of rigid transformations that could be used to show that △ABC≅△EFD. Be sure to explain how you know that each pair of corresponding vertices will overlap perfectly. Then, explain why this sequence illustrates the SAS Congruence Theorem.
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.
To show that △ABC ≅ △EFD using a sequence of rigid transformations, we need to perform a series of movements on △ABC that will make it coincide with △EFD.
Step 1: Translation
First, we can perform a translation to move point A to point E. This means we slide the entire triangle along a straight line without rotating or changing its shape. By doing this, points A and E will overlap perfectly because a translation preserves both distance and angle measurements.
Step 2: Rotation
Next, we need to rotate the triangle to make point B coincide with point F. Since we have already moved point A to E, we can use this as the center of rotation. We rotate the triangle so that line AB coincides with line EF. This rotation will ensure that point B and point F overlap perfectly. A rotation, when performed properly, preserves distance and angle measurements, so the two corresponding sides will remain equal after rotation.
Step 3: Translation
Finally, we need to perform another translation to bring point C to point D. Again, we slide the entire triangle along a straight line without rotating or changing its shape. By doing this, points C and D will overlap perfectly since a translation preserves distances and angles.
By using these sequence of rigid transformations (translation, rotation, and translation), we have shown that all corresponding vertices of △ABC and △EFD overlap perfectly. Therefore, we can conclude that △ABC ≅ △EFD based on the SAS Congruence Theorem.
The SAS Congruence Theorem states that if two triangles have two corresponding sides that are congruent and the included angles between these sides are congruent, then the triangles are congruent. In our case, after the translation, rotation, and translation, we have AB ≅ EF and the included angles at A and B are congruent to the included angles at E and F. Therefore, these transformations illustrate the SAS Congruence Theorem, and we can conclude that △ABC ≅ △EFD.
To show that triangles △ABC and △EFD are congruent, we can use a sequence of rigid transformations. Rigid transformations include translation, rotation, and reflection.
First, let's determine which pair of corresponding vertices we can start with to ensure a perfect overlap. In this case, we know that △ABC and △EFD have corresponding vertices A and E, B and F, and C and D. Let's start with vertices A and E.
To overlap vertices A and E, we can perform a translation. Translation involves sliding an object along a straight line without rotating or changing its shape. We can move triangle △EFD such that vertex E aligns with vertex A. It is important to note that translation preserves both shape and size, ensuring a perfect overlap.
Next, let's overlap vertices B and F. Again, we can perform a translation to slide triangle △EFD such that vertex F aligns with vertex B. Again, translation preserves shape and size, guaranteeing a perfect overlap of the corresponding vertices.
Finally, let's overlap vertices C and D. We can achieve this by performing another translation, sliding triangle △EFD such that vertex D aligns with vertex C. As before, translation preserves shape and size, resulting in a perfect overlap of the corresponding vertices.
Now, let's move on to why this sequence of transformations illustrates the SAS Congruence Theorem. According to the SAS Congruence Theorem, if two sides and the included angle of one triangle are congruent to the corresponding sides and the included angle of another triangle, then the triangles are congruent.
In our case, we have shown that the corresponding sides of triangles △ABC and △EFD are congruent due to the translations. By aligning sides AB and EF, BC and FD, and AC and ED, we ensure that they have the same length, satisfying the S side of the SAS Congruence Theorem.
Additionally, we have also ensured that the included angles A and E, B and F, and C and D are congruent due to the translations. The angles remain the same before and after the translations, satisfying the A angle of the SAS Congruence Theorem.
Therefore, since both the sides and included angles of triangles △ABC and △EFD are congruent, we can conclude that the triangles are congruent, as stated by the SAS Congruence Theorem.