Question

1. below are two triangles that have 2 pairs of corresponding angles congruent, and the corresponding non included sides are congruent. find a sequence of rigid motions that take \\( \triangle a b c \\) onto \\( \triangle x y z \\).

Explanation:

Step1: Translate \(\triangle ABC\)

Translate \(\triangle ABC\) so that point \(A\) maps to point \(X\). Let the translation vector be \(\overrightarrow{AX}\). After translation, \(\triangle ABC\) is moved to a new position where \(A\) coincides with \(X\).

Step2: Rotate the translated triangle

Rotate the translated \(\triangle ABC\) (now with \(A\) at \(X\)) around point \(X\) so that the side (non - included side which was congruent) and the angles match the orientation of \(\triangle XYZ\). Since the two pairs of corresponding angles are congruent and the non - included sides are congruent, after rotation, the two triangles will overlap.

Answer:

First, translate \(\triangle ABC\) so that \(A\) maps to \(X\). Then, rotate the translated \(\triangle ABC\) around \(X\) so that it coincides with \(\triangle XYZ\).