1. **Problem statement:** Four lines intersect at point A forming several angles. Given angles are $4x^\circ$, $(2x+18)^\circ$, and $2x^\circ$. The vertical and horizontal lines are perpendicular, so the angle between them is $90^\circ$. We need to find the value of $x$. 2. **Key fact:** The sum of angles around point A is $360^\circ$. Also, the vertical and horizontal lines form a right angle of $90^\circ$. 3. **Identify angles:** The angles around A are: - $4x^\circ$ (upper-left between rays $A K$ and $A B$) - $(2x+18)^\circ$ (right/below between rays $A D$ and $A E$) - $2x^\circ$ (below between rays $A H$ and $A F$) - $90^\circ$ (right angle between vertical and horizontal lines) 4. **Sum of angles around point A:** $$4x + (2x+18) + 2x + 90 = 360$$ 5. **Simplify the equation:** $$4x + 2x + 18 + 2x + 90 = 360$$ $$ (4x + 2x + 2x) + (18 + 90) = 360$$ $$8x + 108 = 360$$ 6. **Solve for $x$: subtract 108 from both sides:** $$8x + \cancel{108} - \cancel{108} = 360 - 108$$ $$8x = 252$$ 7. **Divide both sides by 8:** $$\frac{8x}{\cancel{8}} = \frac{252}{\cancel{8}}$$ $$x = 31.5$$ **Final answer:** $$\boxed{31.5}$$