[mark all correct answers] \( \square \) a. \( \triangle P Q R \cong \triangle X Y Z \) \( \square \) b. \( \triangle P Q R \cong \triangle X Z Y \) \( \square \) c. \( \triangle P Q R \cong \triangle Y X Z \) \( \square \) d. \( S S S \) \( \square \) e. \( S A S \)

To determine which statements are correct regarding the congruence of triangles, we need to analyze the congruence criteria. The common criteria for triangle congruence include: 1. **Side-Side-Side (SSS)**: If all three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. 2. **Side-Angle-Side (SAS)**: 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. 3. **Angle-Side-Angle (ASA)**: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. 4. **Angle-Angle-Side (AAS)**: If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, the triangles are congruent. 5. **Hypotenuse-Leg (HL)**: For right triangles, if the hypotenuse and one leg of one triangle are equal to the hypotenuse and one leg of another triangle, the triangles are congruent. Now, let's analyze the options provided: - **a. \( \triangle P Q R \cong \triangle X Y Z \)**: This statement suggests that triangle PQR is congruent to triangle XYZ. Without additional information about the sides or angles, we cannot determine if this is true. - **b. \( \triangle P Q R \cong \triangle X Z Y \)**: This statement suggests that triangle PQR is congruent to triangle XZY. Again, without additional information, we cannot determine if this is true. - **c. \( \triangle P Q R \cong \triangle Y X Z \)**: This statement suggests that triangle PQR is congruent to triangle YXZ. Without additional information, we cannot determine if this is true. - **d. \( S S S \)**: This indicates the Side-Side-Side congruence criterion. If the sides of the triangles are equal, then this statement is true. - **e. \( S A S \)**: This indicates the Side-Angle-Side congruence criterion. If two sides and the included angle are equal, then this statement is true. Since we do not have specific information about the triangles' sides or angles, we can only confirm the validity of the congruence criteria mentioned in options d and e. Thus, the correct answers are: - \( \square \) d. \( S S S \) - \( \square \) e. \( S A S \)