Question

with the information given in the diagram, which statement is not true?
rigid motions can be used to match the triangles up exactly.
the triangles can be proved by the sss theorem to be congruent.
the triangles can be proved by the aas theorem to be congruent.
the triangles can be proved by the sas theorem to be congruent.

Explanation:

Step1: Analyze the SSS (Side - Side - Side) theorem

The SSS theorem states that if three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. In the given diagram, we have two triangles \( \triangle YGN\) and \( \triangle QGN\). We know that \( YN = QN\), \( YG=QG\), and \( GN = GN\) (common side). So, by SSS, \( \triangle YGN\cong\triangle QGN\).

Step2: Analyze the SAS (Side - Angle - Side) theorem

The SAS theorem states that 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. We have \( YN = QN\), \( YG = QG\), and \( \angle YNG+\angle QNG = 180^{\circ}\) (linear pair). But if we consider the non - linear pair angles, we can also show congruence using SAS.

Step3: Analyze the rigid motion

Rigid motions (translation, rotation, reflection) can be used to map one triangle onto the other since the triangles are congruent.

Step4: Analyze the AAS (Angle - Angle - Side) theorem

The AAS theorem states that if two angles and a non - included side of one triangle are equal to two angles and the corresponding non - included side of another triangle, the triangles are congruent. There is no information given about two angles in the problem statement to apply the AAS theorem.

Answer:

The triangles can be proved by the AAS theorem to be congruent.