Question

use the image to answer the question.

describe a sequence of rigid transformations that could be used to show that (\triangle abccong\triangle 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.

(4 points)

Explanation:

Step1: Translation

Translate \(\triangle ABC\) so that point \(B\) maps to point \(F\). Since \(AB = FD\) (marked as equal), after translation, \(A\) will be in line with \(D\) as the length is same.

Step2: Rotation

Rotate \(\triangle ABC\) (after translation) around point \(F\) (which was \(B\)) so that \(BC\) aligns with \(FE\). Since \(\angle ABC=\angle EFD\) (marked as equal), the rotation will make the angles match. And as \(BC = FE\) (marked as equal), the sides will overlap.

For SAS Congruence: We have two sides (\(AB = FD\), \(BC = FE\)) and the included angle (\(\angle ABC=\angle EFD\)) equal. Rigid transformations (translation and rotation) preserve side - lengths and angles. So, when we perform these transformations, the two triangles overlap exactly, which is what the SAS Congruence Theorem (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) states.

Answer:

First, translate \(\triangle ABC\) so that \(B\) maps to \(F\). Then rotate \(\triangle ABC\) (after translation) around \(F\) so that \(BC\) aligns with \(FE\). The translation works because \(AB = FD\) (ensures \(A\) maps to \(D\) after translation of \(B\) to \(F\)) and the rotation works because \(\angle ABC=\angle EFD\) (makes the angle match) and \(BC = FE\) (makes the side match). This sequence illustrates SAS as we have two sides (\(AB = FD\), \(BC = FE\)) and the included angle (\(\angle ABC=\angle EFD\)) congruent, and rigid transformations (which preserve side - lengths and angles) show the triangles overlap (i.e., are congruent) as per the SAS Congruence Theorem.