Triangle Similarity 791169

1. **Problem Statement:** Prove that triangle HEA is similar to triangle HTA given quadrilateral MATH with conditions: HM \cong AT, HT \cong AM, HE \perp MEA, and HA \perp AT. 2. **Identify the triangles and angles:** We want to prove similarity between \triangle HEA and \triangle HTA. 3. **Given perpendicularities:** HE \perp MEA means \angle HEA = 90^\circ. Also, HA \perp AT means \angle HAT = 90^\circ. 4. **Right angles in both triangles:** Both \triangle HEA and \triangle HTA have a right angle at E and A respectively. 5. **Common angle:** Both triangles share \angle A (vertex A). 6. **Angle-Angle (AA) similarity criterion:** Two triangles are similar if they have two corresponding angles equal. Here, \angle HEA = \angle HTA = 90^\circ and \angle EAH = \angle TAH (common angle). 7. **Conclusion:** By AA similarity criterion, \triangle HEA \sim \triangle HTA. **Final answer:** \boxed{\triangle HEA \sim \triangle HTA}