Parallel Lines X 325F55

1. **State the problem:** We are given two parallel lines $m$ and $n$ cut by a transversal, creating two angles: $(2x+22)^\circ$ and $(4x+2)^\circ$. We need to find the value of $x$. 2. **Identify the relationship between the angles:** Since $m \parallel n$ and the transversal cuts them, the angles $(2x+22)^\circ$ and $(4x+2)^\circ$ are alternate interior angles. 3. **Use the property of alternate interior angles:** Alternate interior angles are equal when lines are parallel. $$2x + 22 = 4x + 2$$ 4. **Solve the equation:** $$2x + 22 = 4x + 2$$ Subtract $2x$ from both sides: $$\cancel{2x} + 22 = \cancel{2x} + 4x + 2 \implies 22 = 2x + 2$$ Subtract $2$ from both sides: $$22 - 2 = 2x + \cancel{2} - \cancel{2} \implies 20 = 2x$$ Divide both sides by $2$: $$\frac{20}{\cancel{2}} = \frac{2x}{\cancel{2}} \implies 10 = x$$ 5. **Final answer:** $$\boxed{10}$$