Question

is \\( \triangle def \\) congruent to \\( \triangle xyz \\)? justify your answer using rigid motions.

yes. a translation can map \\( \triangle def \\) onto \\( \triangle xyz \\).

yes. a reflection can map \\( \triangle def \\) onto \\( \triangle xyz \\).

no. it is not possible to map \\( \triangle def \\) onto \\( \triangle xyz \\) using only rigid motions.

Explanation:

Step1: Analyze the triangles

In $\triangle DEF$, the side - angle - side (SAS) relationship: we know the angles and side lengths. Rigid motions (translation, rotation, reflection) preserve the shape and size of a figure.

Step2: Check the correspondence

For a translation, we move the figure without rotation or reflection. If we try to map $\triangle DEF$ to $\triangle XYZ$ by translation, we need to check the correspondence of vertices. The vertex \(D\) with \(80^{\circ}\) angle and \(F\) with \(75^{\circ}\) angle. In $\triangle XYZ$, vertex \(X\) has \(80^{\circ}\) and \(Z\) has \(75^{\circ}\). But the side lengths adjacent to the angles (indicated by the tick - marks) do not correspond in the order required for a translation.
For a reflection, we flip the figure. But the side - angle - side correspondence (based on the tick - marks for side lengths and angle measures) still does not hold.

Answer:

No. It is not possible to map $\triangle DEF$ onto $\triangle XYZ$ using only rigid motions.