Question

is △abc congruent to △xyz? justify your answer using rigid motions.
yes. a translation and a rotation can map △abc onto △xyz.
yes. a translation and a reflection can map △abc onto △xyz.
no. it is not possible to map △abc onto △xyz using only rigid motions.

Explanation:

Step1: Identify side - lengths

In $\triangle ABC$, side - lengths are $AB = 4$ in, $BC = 3$ in, $AC = 6$ in. In $\triangle XYZ$, side - lengths are $XY = 4$ in, $YZ = 3$ in, $XZ = 6$ in.

Step2: Recall congruence by SSS

Since $AB = XY$, $BC = YZ$, and $AC = XZ$, by the Side - Side - Side (SSS) congruence criterion, $\triangle ABC\cong\triangle XYZ$. Also, a translation can be used to move $\triangle ABC$ so that one of its vertices coincides with a corresponding vertex of $\triangle XYZ$, and then a rotation can be used to align the other sides.

Answer:

Yes. A translation and a rotation can map $\triangle ABC$ onto $\triangle XYZ$.