Determining Congruence

The triangles have different orientation and location. In $△XYZ,$ the vertex $Z$ is to the right of $X$ and $Y.$ In $△X^{\prime }{Y}^{\prime }{Z}^{\prime },$ the corresponding vertex $Z^{\prime },$ is to the left of $X^{\prime }$ and $Y^{\prime }.$ To line up their orientations, we can rotate $△XYZ$ $180^{\circ }$ about the origin.

Now, by translating the rotated triangle down by $1$ unit and left by 5 units, we can map it onto $△X^{\prime }{Y}^{\prime }{Z}^{\prime }.$

Therefore, the congruence transformation that maps $△XYZ$ onto $△X^{\prime }{Y}^{\prime }{Z}^{\prime }$ can be written as follows. $$\begin{array}{rl}\text{Rotation: }& 180^{\circ }\text{ rotation about the origin}\\ \text{Translation: }& (x,y)\to (x-5,y-1).\end{array}$$