Question

which rule explains why these triangles are congruent?
aas
sas
asa
sss
these triangles cannot be proven congruent.

Explanation:

Step1: Recall congruence rules

• AAS (Angle - Angle - Side): Two angles and a non - included side of one triangle are equal to the corresponding two angles and non - included side of another triangle.
• SAS (Side - Angle - Side): Two sides and the included angle of one triangle are equal to the corresponding two sides and included angle of another triangle.
• ASA (Angle - Side - Angle): Two angles and the included side of one triangle are equal to the corresponding two angles and included side of another triangle.
• SSS (Side - Side - Side): All three sides of one triangle are equal to the corresponding three sides of another triangle.

Step2: Analyze the given triangles

• In \(\triangle PQN\) and \(\triangle SQN\):
• We have \(PQ = SQ\) (marked as equal sides).
• \(\angle P=\angle S\) (marked as equal angles).
• \(QN = QN\) (common side, which is a non - included side with respect to the given angles \(\angle P\) and \(\angle S\)).

Step3: Apply the AAS rule

Since we have two angles (\(\angle P\) and \(\angle S\)) and a non - included side (\(QN\)) equal in the two triangles \(\triangle PQN\) and \(\triangle SQN\), the AAS (Angle - Angle - Side) congruence rule applies.

Answer:

AAS