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.
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Sequence of Rigid Transformations:
1. Translation: Move triangle EFD (the second triangle) so that point E coincides with point A in triangle ABC (the first triangle).
Explanation: We can slide triangle EFD horizontally or vertically without changing its shape in order to align point E with point A. This move ensures that the corresponding vertices E and A of both triangles will overlap perfectly.
2. Rotation: Rotate triangle EFD around point E until side EF aligns with side AB.
Explanation: By rotating triangle EFD around point E, we can bring side EF into the same position as side AB. This rotation ensures that the corresponding sides EF and AB of both triangles will overlap perfectly.
3. Translation: Move triangle EFD (the rotated triangle) so that point D coincides with point C in triangle ABC (the first triangle).
Explanation: We can slide triangle EFD horizontally or vertically without changing its shape in order to align point D with point C. This move ensures that the corresponding vertices D and C of both triangles will overlap perfectly.
This sequence of rigid transformations illustrates the SAS Congruence Theorem. According to this theorem, if two triangles have two pairs of corresponding sides that are congruent and the angles between these sides are also congruent, then the triangles are congruent.
In our case, we have shown that side AB is congruent to side EF, side BC is congruent to side FD, and angle BAC is congruent to angle EFD. These are the pairs of corresponding sides and angles that satisfy the SAS Congruence Theorem. Therefore, by performing these rigid transformations, we have proven that triangle ABC is congruent to triangle EFD.
To show that △ABC is congruent to △EFD, we can use a sequence of rigid transformations. Rigid transformations are movements of an object that do not change its size or shape.
Step 1: Translation
First, we can translate triangle △EFD so that the point E coincides with point A. This means we move the entire triangle in a straight line without rotating or changing its size. By doing this, point E and point A will overlap perfectly.
Step 2: Rotation
Next, we can rotate triangle △EFD around point A so that line segment EF coincides with line segment AB. This means we turn the triangle around point A while keeping all the angles the same. By doing this, line segment EF and line segment AB will be in the same position and direction, overlapping perfectly.
Step 3: Translation
Finally, we can translate triangle △EFD so that the point D coincides with point C. This means we move the entire triangle in a straight line without rotating or changing its size. By doing this, point D and point C will overlap perfectly.
By performing these sequence of rigid transformations, we have shown that each pair of corresponding vertices (A and E, B and F, C and D) overlap perfectly. This means that the triangles △ABC and △EFD are congruent.
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 included angle of another triangle, then the two triangles are congruent. In our sequence, we have shown that the sides AB and EF are congruent, and the angles at A and E are congruent (as they overlap perfectly). 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 are transformations that do not change the shape or size of an object. The transformations we will use are translation, rotation, and reflection.
First, we can start by translating triangle ABC to match triangle EFD. Translation is simply moving an object without changing its shape or size. By sliding triangle ABC in a specific direction, we can align its corresponding vertices with those of triangle EFD.
Next, we can rotate triangle ABC so that it matches the orientation of triangle EFD. Rotation is like turning an object around a fixed point. By rotating triangle ABC in a specific direction and by a specific angle, we can make sure that its corresponding sides and angles align perfectly with those of triangle EFD.
Finally, we can reflect triangle ABC to make it exactly match triangle EFD. Reflection is like flipping an object over a line. By reflecting triangle ABC across a specific line, we can ensure that its corresponding sides and angles overlap perfectly with those of triangle EFD.
We know that each pair of corresponding vertices will overlap perfectly because a rigid transformation guarantees that the shape and size of the object remain unchanged. This means that if we perform the same transformation on two different triangles, and the corresponding sides and angles of the triangles align, then the triangles are congruent.
This sequence of rigid transformations illustrates the 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 case, by using translation, rotation, and reflection to make triangle ABC congruent to triangle EFD, we are showing that the sides and the included angle of triangle ABC are congruent to the sides and the included angle of triangle EFD. Therefore, according to the SAS Congruence Theorem, triangle ABC is congruent to triangle EFD.