Question

is \\( \triangle wxy \\) congruent to \\( \triangle jkl \\)? justify your answer using rigid motions.
yes. a rotation and a dilation can map \\( \triangle wxy \\) onto \\( \triangle jkl \\).
yes. a translation and a rotation can map \\( \triangle wxy \\) onto \\( \triangle jkl \\).
no. it is not possible to map \\( \triangle wxy \\) onto \\( \triangle jkl \\) using only rigid motions.

Explanation:

Step1: Recall the definition of rigid motions

Rigid motions (isometries) include translations, rotations, and reflections. These motions preserve the shape and size of a figure.

Step2: Check the side - length and angle measures

In \(\triangle WXY\) and \(\triangle JKL\), the side - lengths \(WX = JL=7\) in, \(XY = KL = 4\) in, \(WY=JL = 6\) in. The angle measures: \(\angle WXY=\angle JKL = 58.8^{\circ}\), \(\angle XYW=\angle KLJ=86.4^{\circ}\), \(\angle YWX=\angle LJK = 34.8^{\circ}\).

Step3: Analyze the effect of rigid motions

A translation (sliding) and a rotation (turning) can be used to map one triangle onto the other. A translation moves \(\triangle WXY\) without changing its orientation or size, and a rotation can then be used to align the triangles properly. Dilation is not a rigid motion (it changes the size of a figure).

Answer:

Yes. A translation and a rotation can map \(\triangle WXY\) onto \(\triangle JKL\).