Question

use the image to answer the question. describe a sequence of rigid transformations that could be used to show that ( \triangle abc cong \triangle def ). explain how you know each transformation will result in a congruent figure. then, explain how this sequence of rigid transformations illustrates the asa congruence theorem. (4 points)

Explanation:

Step1: Translation

Translate \(\triangle ABC\) so that point \(A\) maps to point \(D\). A translation is a rigid transformation. It preserves the shape and size of the figure. So, \(\triangle ABC\) and its image after translation are congruent.

Step2: Rotation

Rotate the translated \(\triangle ABC\) (now with \(A\) at \(D\)) around point \(D\) so that side \(AC\) (now \(DC'\)) maps to side \(DF\). A rotation is a rigid transformation. It preserves the shape and size of the figure. So, the figure after rotation is congruent to the previous image (which was congruent to \(\triangle ABC\)).

Step3: ASA Congruence

We know from the problem that \(\angle C=\angle D\), \(AC = DF\), and \(\angle A=\angle F\) (by the rigid - transformations, corresponding angles are equal). In the ASA (Angle - Side - Angle) Congruence Theorem, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. Here, after the rigid transformations, we have shown that \(\angle C\cong\angle D\), \(AC\cong DF\), \(\angle A\cong\angle F\) (the included side \(AC/DF\) between \(\angle C\) and \(\angle A\) in \(\triangle ABC\) and between \(\angle D\) and \(\angle F\) in \(\triangle DEF\)).

Answer:

First, translate \(\triangle ABC\) so that \(A\) maps to \(D\) (translation preserves congruence). Then, rotate the translated triangle around \(D\) so that \(AC\) maps to \(DF\) (rotation preserves congruence). For ASA: After transformations, \(\angle C\cong\angle D\), \(AC\cong DF\), \(\angle A\cong\angle F\) (by rigid - transformation properties), satisfying the ASA Congruence Theorem (\(two\ angles\) (\(\angle C,\angle A\) and \(\angle D,\angle F\)) \(and\ the\ included\ side\) (\(AC = DF\)) are congruent).