Question

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

Explanation:

Step1: Analyze the given information

We have two triangles \(\triangle TUQ\) and \(\triangle TSQ\). We know that \(TU = TS\) (marked with one - tick), \(\angle U=\angle S\) (marked angles), and \(TQ=TQ\) (common side).

Step2: Recall the congruence rules

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

In our case, we have two angles (\(\angle U\) and \(\angle TQU=\angle TQS\) (vertical angles are equal)) and a non - included side (\(TQ\)). But we can also use the fact that we have two angles (\(\angle U=\angle S\)) and a non - included side (\(TU = TS\)).

By the AAS (Angle - Angle - Side) congruence rule, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Answer:

AAS