1. **State the problem:** We are given two parallel lines cut by a transversal, with angles labeled. We know $m\angle 2 = 4x - 5$ and $m\angle 6 = 2x + 17$. We need to find $m\angle 6$. 2. **Identify the relationship:** Since lines $a$ and $b$ are parallel and cut by a transversal, angles 2 and 6 are alternate interior angles, so they are equal. 3. **Set up the equation:** $$4x - 5 = 2x + 17$$ 4. **Solve for $x$:** $$4x - 5 = 2x + 17$$ $$4x - \cancel{5} - 2x = 2x + 17 - \cancel{5}$$ $$2x = 22$$ $$x = \frac{22}{2}$$ $$x = 11$$ 5. **Find $m\angle 6$ by substituting $x=11$ into $2x + 17$:** $$m\angle 6 = 2(11) + 17 = 22 + 17 = 39$$ 6. **Answer:** $m\angle 6 = 39^\circ$