An illustration shows two triangles side by side. The two triangles are labeled A B C and D E F. Side A B is congruent to side E F. Side B C is congruent to side F D. Angle B is congruent to angle F.
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 can use the following sequence of rigid transformations:
1. Translation: Move triangle △ABC so that vertex A coincides with vertex E.
By translating triangle △ABC, we are shifting the entire triangle without changing its shape or size. Since we know that side AB is congruent to side EF, and angle B is congruent to angle F, this translation ensures that side AB aligns perfectly with side EF, and angle B overlaps with angle F.
2. Rotation: Rotate triangle △ABC so that side BC aligns with side FD.
By rotating triangle △ABC, we are turning it around a fixed point (in this case, vertex E) without altering its shape or size. Since we know that side BC is congruent to side FD, this rotation ensures that side BC overlaps with side FD, without changing the alignment of side AB with side EF or angle B with angle F.
After the translation and rotation, triangle △ABC is now superimposed on triangle △EFD, meaning that all corresponding vertices (A and E, B and F, C and D) are in the exact same position.
This sequence of rigid transformations illustrates the SAS Congruence Theorem because it shows that two triangles are congruent if they have two pairs of corresponding sides that are congruent and a pair of corresponding angles that are congruent. In this case, we have side AB congruent to side EF, side BC congruent to side FD, and angle B congruent to angle F, satisfying the requirements of the SAS Congruence Theorem. Therefore, we can conclude that △ABC≅△EFD.