1. **Problem Statement:** We are given that $\angle B \cong \angle Y$, $AB \cong XY$, and $BC \cong YZ$. We need to determine which triangle congruency theorem proves the two triangles congruent. 2. **Recall Triangle Congruency Theorems:** - **Angle-Side-Angle (ASA):** Two angles and the included side are congruent. - **Side-Angle-Side (SAS):** Two sides and the included angle are congruent. - **Angle-Angle-Side (AAS):** Two angles and a non-included side are congruent. - **Side-Side-Side (SSS):** All three sides are congruent. 3. **Analyze Given Information:** - $\angle B \cong \angle Y$ (angle) - $AB \cong XY$ (side) - $BC \cong YZ$ (side) 4. **Check if the angle is included between the two sides:** - The angle $B$ is between sides $AB$ and $BC$ in triangle $ABC$. - The angle $Y$ is between sides $XY$ and $YZ$ in triangle $XYZ$. 5. **Conclusion:** Since two sides and the included angle are congruent, the **Side-Angle-Side (SAS)** theorem applies. 6. **Triangle Congruency Statement:** $$\text{cong}(\triangle ABC, \triangle XYZ)$$ **Final answer:** The Side-Angle-Side Triangle Congruency Theorem proves the triangles congruent.