1. **State the problem:** We are given that $ST \cong SN$ and $\angle 1 \cong \angle 2$. We need to prove that $\triangle STY \cong \triangle SNX$. 2. **Identify given information and what to prove:** - Given: $ST = SN$ (sides are congruent) - Given: $\angle 1 = \angle 2$ (angles are congruent) - To prove: $\triangle STY \cong \triangle SNX$ 3. **Look for a congruence postulate:** To prove two triangles congruent, we can use Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), or Angle-Angle-Side (AAS). 4. **Analyze the triangles:** - Both triangles share side $SY$ (or $SX$ depending on labeling), but since the problem states $ST \cong SN$, and angles $\angle 1 \cong \angle 2$, we need to find the third pair of congruent parts. 5. **Use the shared side:** The triangles share side $SY$ or $SX$ (depending on the diagram), so $SY = SX$. 6. **Apply SAS postulate:** We have two sides and the included angle congruent: $$ST = SN, \quad \angle 1 = \angle 2, \quad SY = SX$$ Therefore, by SAS, $\triangle STY \cong \triangle SNX$. 7. **Conclusion:** The triangles are congruent by the SAS postulate. Final answer: $$\triangle STY \cong \triangle SNX$$