1. **State the problem:** Prove that triangles $\triangle VYZ$ and $\triangle YWX$ are similar given that $\angle VYZ = \angle YWX$, $HL = HE = LE$, and $m\angle VYZ + m\angle ZZYX = 180^\circ$. 2. **Recall similarity criteria:** Two triangles are similar if they have two pairs of corresponding angles equal (AA criterion). 3. **Given:** - $\angle VYZ = \angle YWX$ (given) - $HL = HE = LE$ implies certain equal segments that help establish angle relationships. - $m\angle VYZ + m\angle ZZYX = 180^\circ$ means $\angle VYZ$ and $\angle ZZYX$ are supplementary. 4. **Analyze angles:** Since $\angle VYZ$ and $\angle YWX$ are equal, and $\angle VYZ + \angle ZZYX = 180^\circ$, it follows that $\angle ZZYX$ is supplementary to $\angle YWX$. 5. **Identify corresponding angles:** Because $HL = HE = LE$, the triangles share proportional sides and equal angles at $Y$ and $W$. 6. **Conclude similarity:** By AA criterion, since two pairs of angles are equal, $\triangle VYZ \sim \triangle YWX$. **Final answer:** $$\triangle VYZ \sim \triangle YWX$$