1. The problem asks to identify the theorem used to get the third reason in proving $\triangle STY \cong \triangle SNX$ given $ST \cong SN$ and $\angle 1 \cong \angle 2$. 2. In triangle congruence proofs, the third reason often involves showing a third pair of sides or angles are congruent to apply a congruence postulate like SAS, ASA, or SSS. 3. Since $ST \cong SN$ is given and $\angle 1 \cong \angle 2$ is given, the third reason likely involves showing $SY \cong SX$. 4. The theorem used to establish this third congruence is often the Reflexive Property of Congruence if $SY$ and $SX$ are the same segment or a property of corresponding parts of congruent triangles (CPCTC) if previously proven. 5. Therefore, the third reason is typically justified by the Reflexive Property of Congruence or CPCTC, depending on the context. Final answer: The theorem used for the third reason is the Reflexive Property of Congruence or CPCTC.