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 and △EFD are congruent using rigid transformations, we need to find a sequence of transformations such that the corresponding vertices of the two triangles overlap perfectly.
1. First, we can perform a translation. We can move triangle △EFD so that point E coincides with point C. This translation ensures that the corresponding sides are parallel and have the same length.
2. Next, we can perform a rotation. We rotate triangle △EFD around point E so that side ED aligns with side BC. This rotation ensures that the corresponding angles at E and C are congruent.
3. Finally, we can perform another translation. We move triangle △EFD so that point F coincides with point B. This translation ensures that the corresponding sides are parallel and have the same length.
By carefully performing these transformations, we can ensure that all three pairs of corresponding vertices, A and D, B and E, and C and F, overlap perfectly.
This sequence illustrates the SAS Congruence Theorem. SAS stands for Side-Angle-Side, which states that if two triangles have two pairs of corresponding sides that are congruent, and the included angle between these sides is also congruent, then the triangles are congruent.
In our case, the sequence of transformations ensures that side AB is congruent to side DE, side BC is congruent to side EF, and angle ABC is congruent to angle DEF. Therefore, by the SAS Congruence Theorem, we can conclude that △ABC is congruent to △EFD.
To show that △ABC is congruent to △EFD, we can use a sequence of rigid transformations. Rigid transformations preserve both shape and size, so if we can map △ABC onto △EFD using only rigid transformations, it will imply that the two triangles are congruent.
Here is a step-by-step description of the sequence of rigid transformations that ensures the two triangles overlap perfectly:
1. Translation: We can start by performing a translation to move point A to point E. By sliding the entire triangle ABC to the right or left, we can ensure that A coincides with E.
2. Rotation: After the translation, we can perform a rotation about point E to align segment EF with segment AB. This rotation will bring vertex B to the position of vertex C and vertex C to the position of vertex D.
3. Translation: Finally, we can perform another translation to move line ED to line BC. This translation will bring point D to the position of point B, resulting in the overlap of all corresponding vertices.
By following this sequence of rigid transformations, we ensure that each pair of corresponding vertices overlaps perfectly. Vertex A is moved to E through translation, then vertices B and C are aligned with D through rotation, and finally, D is moved to B through another translation.
This sequence illustrates the SAS Congruence Theorem, which states that if two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, then the two triangles are congruent. In our case, segment AB is congruent to segment EF, segment BC is congruent to segment DE, and angle ABC is congruent to angle EFD due to the rigid transformations. Thus, we can conclude that △ABC is congruent to △EFD by applying the SAS Congruence Theorem.
To show that triangles △ABC and △EFD are congruent, we can use a sequence of rigid transformations. Rigid transformations preserve shape and size, meaning that if two shapes can be superimposed using rigid motions, they are congruent.
Here's a possible sequence of rigid transformations to show the congruence:
1. Translation: Move △ABC to a new location so that one of its vertices coincides with a corresponding vertex of △EFD. This can be done by sliding △ABC without rotating or changing its size. By doing this, we ensure that the corresponding vertices overlap perfectly.
2. Rotation: Rotate △ABC about the coinciding vertex until another pair of corresponding vertices coincide. This rotation should be done in such a way that the angle of rotation is equal to the angle between the corresponding sides of the two triangles. This step guarantees that the corresponding sides are in alignment.
3. Translation: Finally, translate △ABC one more time without rotation so that the remaining pair of corresponding vertices coincide. Again, make sure the translation is such that the sides coincide properly.
Now, let's explain why this sequence of transformations illustrates the SAS (Side-Angle-Side) Congruence Theorem:
First, by the definition of congruence, the figures must have the same shape and size. Each transformation we performed, including the translations and the rotation, individually preserves the shape and size of the triangle.
Second, the translation ensures that one pair of corresponding sides of the triangles coincide, specifically, the two sides connected to the first vertex. The rotation ensures that the angle between these sides is the same in both triangles. Finally, the last translation guarantees that the remaining pair of corresponding sides coincide.
By meeting the requirements for SAS congruence (matching two pairs of sides and the included angle), we have shown that triangles △ABC and △EFD are congruent. Therefore, the sequence of transformations we used demonstrates the SAS Congruence Theorem.