1. The problem asks to identify the theorem used as the third reason in proving $\triangle STY \cong \triangle SNX$ given $ST \cong SN$ and $\angle 1 \cong \angle 2$. 2. The first two reasons are $ST \cong SN$ (given) and $\angle 1 \cong \angle 2$ (given). 3. To prove triangle congruence, a third pair of corresponding parts must be congruent. 4. The third reason is often $SY \cong SX$. 5. The theorem used to justify $SY \cong SX$ is the Side-Side-Side (SSS) Congruence Postulate if all three sides are congruent, or the Side-Angle-Side (SAS) Postulate if two sides and the included angle are congruent. 6. Since $ST \cong SN$ and $\angle 1 \cong \angle 2$ are given, the third reason is $SY \cong SX$ by the Reflexive Property of Congruence because $SY$ and $SX$ are the same segment shared by both triangles. 7. Therefore, the third reason is justified by the Reflexive Property of Congruence. Final answer: The theorem used for the third reason is the Reflexive Property of Congruence.