Question

here are 3 triangles. 1. triangle pqr is congruent to which triangle? explain your reasoning. 2. show a sequence of rigid motions that takes triangle pqr to that triangle. draw each step of the transformation. 3. explain why there cant be a rigid motion from triangle pqr to the other triangle.

Explanation:

Step1: Identify congruence criteria

In $\triangle PQR$ and $\triangle NML$, we have $\angle P = 46.6^{\circ}$, $PQ = 5.9$, $QR=4.3$. In $\triangle NML$, $\angle M = 46.6^{\circ}$, $MN = 5.9$, $ML = 4.3$. By the Side - Angle - Side (SAS) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the two triangles are congruent.

Step2: Describe rigid - motion sequence

First, translate $\triangle PQR$ so that point $P$ coincides with point $M$. Then, rotate $\triangle PQR$ about the new - position of point $P$ (which is now at $M$) so that side $PQ$ coincides with side $MN$. Since the angles and side - lengths match, $\triangle PQR$ will coincide with $\triangle NML$.

Step3: Explain non - congruence

For $\triangle PQR$ and $\triangle ACE$, although $\angle P=\angle A = 46.6^{\circ}$ and two side - lengths are the same ($4.3$ and $5.9$), the side - angle - side relationship is not the same. In $\triangle PQR$, the $46.6^{\circ}$ angle is between sides of lengths $5.9$ and $4.3$, while in $\triangle ACE$, the $46.6^{\circ}$ angle is not between the sides of lengths $5.9$ and $4.3$. So, there is no rigid motion that can map $\triangle PQR$ to $\triangle ACE$.

Answer:

1. $\triangle PQR$ is congruent to $\triangle NML$ by the SAS congruence criterion.
2. First, translate $\triangle PQR$ so that $P$ coincides with $M$. Then rotate $\triangle PQR$ about the new - position of $P$ (at $M$) so that $PQ$ coincides with $MN$.
3. For $\triangle PQR$ and $\triangle ACE$, the side - angle - side relationships do not match, so there is no rigid motion from $\triangle PQR$ to $\triangle ACE$.