1. **State the problem:** Given that \(\overline{TV}\) bisects \(\angle SYU\) and \(\angle TSY \cong \angle TUV\), prove that \(\triangle TSY \cong \triangle TUV\). 2. **Identify given information and what to prove:** - \(\overline{TV}\) bisects \(\angle SYU\) means \(\angle STV \cong \angle YTV\). - \(\angle TSY \cong \angle TUV\) is given. - We want to prove \(\triangle TSY \cong \triangle TUV\). 3. **Use the Angle-Angle-Side (AAS) congruence theorem:** - AAS states that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent. 4. **Show the pairs of congruent parts:** - \(\angle TSY \cong \angle TUV\) (given). - \(\angle STV \cong \angle YTV\) (since \(\overline{TV}\) bisects \(\angle SYU\)). - Side \(\overline{TV}\) is common to both triangles (shared side). 5. **Write the congruence statement:** - By AAS, \(\triangle TSY \cong \triangle TUV\). \[\boxed{\triangle TSY \cong \triangle TUV}\]