1. **State the problem:** We have two triangles, VWY and UYX, with VW parallel to UX. Given lengths UV = 10, UY = 20, and YX = 44, we need to find WX. 2. **Use the properties of parallel lines and similar triangles:** Since VW \parallel UX, triangles VWY and UYX are similar by the AA criterion. 3. **Set up the ratio of corresponding sides:** The ratio of sides in similar triangles is equal. So, $$\frac{VW}{UX} = \frac{VY}{UY} = \frac{WY}{YX}$$ 4. **Express WX in terms of known lengths:** We want WX, which is part of segment UX. Since UX = UY + YX = 20 + 44 = 64, and VW is parallel to UX, WX corresponds to WY in the smaller triangle. 5. **Find the ratio of sides:** Since UV = 10 and UY = 20, the ratio of UV to UY is $$\frac{UV}{UY} = \frac{10}{20} = \frac{1}{2}$$ 6. **Use the ratio to find WX:** Because triangles are similar, WX corresponds to half of YX: $$WX = \frac{1}{2} \times 44 = 22$$ **Final answer:** $$\boxed{22}$$