Question

which of the following statements are true about the triangles?
the triangles are not congruent.
the triangles are congruent, because of the asa theorem.
the triangles are congruent, because of the sas theorem.
the triangles are congruent because of the sss theorem

Explanation:

Step1: Analyze the given information

We have two triangles \(\triangle XYZ\) and \(\triangle PQR\). We know that one angle is equal (the marked angles at \(X\) and \(P\)), and two pairs of sides are equal (the sides with one - cross and two - cross marks).

Step2: Recall the congruence theorems

• 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.
• 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.
• SSS (Side - Side - Side): All three sides of one triangle are equal to all three sides of another triangle.

In our case, we have two sides and the included angle (the marked angle) equal.

Answer:

The triangles are congruent, because of the SAS Theorem.