To determine if triangles \( \triangle QRS \) and \( \triangle TUV \) are congruent when \( x = -4 \), we need to analyze the given side lengths or angles of the triangles using the appropriate congruence theorem.
However, without specific measurements or relationships between the angles and sides of triangles, we cannot definitively determine their congruence.
To analyze congruence, you can apply the following rules:
- SSS (Side-Side-Side) Congruence Theorem: If all three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
- SAS (Side-Angle-Side) Congruence Theorem: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
- ASA (Angle-Side-Angle) Congruence Theorem: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.
- AAS (Angle-Angle-Side) Congruence Theorem: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.
- HL (Hypotenuse-Leg) Congruence Theorem: For right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another triangle, then the triangles are congruent.
Since the solution options you provided include specific congruence theorems and a response about non-congruence, you must check the specific measurements of \( \triangle QRS \) and \( \triangle TUV \) with \( x = -4 \) to see which theorem applies.
If you have specific numerical side lengths or angle measures for the triangles with \( x = -4 \), please provide those to resolve the question accurately. If the details indicate they can be congruent via HL, SAS, SSS or any other theorem, then that would define your congruence statement. If they cannot be confirmed as congruent, the correct conclusion would be that "the triangles are not congruent."