1. **State the problem:** Given that $\angle M \cong \angle T$ and $\angle MAH \cong \angle THA$, prove that segment $HM \cong AT$. 2. **List the given information:** - $\angle M \cong \angle T$ (Given) - $\angle MAH \cong \angle THA$ (Given) 3. **Identify the reflexive property:** - Segment $AH$ is common to both triangles $MAH$ and $THA$, so $AH \cong AH$ by the Reflexive Property. 4. **Apply the Angle-Angle-Side (AAS) Congruence Theorem:** - Triangles $MAH$ and $THA$ have two pairs of congruent angles and the included side $AH$ congruent. - Therefore, $\triangle MAH \cong \triangle THA$ by AAS. 5. **Conclude the proof:** - Corresponding parts of congruent triangles are congruent (CPCTC). - Hence, $HM \cong AT$. Final answer: $HM \cong AT$