Question

activity 1: which triangles are congruent?
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: Check congruence using SAS

In triangle \(PQR\) and triangle \(NML\), \(\angle P = \angle M=66.6^{\circ}\), \(PQ = ML = 5.9\), \(PR = MN = 4.3\). 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 triangles are congruent.

Step2: Rigid motion sequence

First, rotate triangle \(PQR\) so that the orientation of \(\angle P\) matches the orientation of \(\angle M\). Then, translate triangle \(PQR\) so that point \(P\) maps to point \(M\), point \(Q\) maps to point \(L\) and point \(R\) maps to point \(N\).

Step3: Analyze non - congruent triangle

For triangle \(AEC\), in triangle \(PQR\) the sides adjacent to the \(66.6^{\circ}\) angle are \(4.3\) and \(5.9\). In triangle \(AEC\), the side adjacent to the \(66.6^{\circ}\) angle is \(4.3\) and \(5.9\) is not adjacent to the \(66.6^{\circ}\) angle. So, there is no rigid motion (since rigid motions preserve side - angle - side relationships) that can map \(PQR\) to \(AEC\)

Answer:

1. Triangle \(PQR\) is congruent to triangle \(NML\) because of the SAS (Side - Angle - Side) congruence criterion.
2. First, rotate triangle \(PQR\) to match the angle orientation. Then, translate it to align the vertices.
3. In triangle \(AEC\), the side - angle - side relationship does not match that of triangle \(PQR\), so no rigid motion exists.