1. **Stating the problem:** We are given three angles: $ (4x + 15)^\circ $, $ (30x - 5)^\circ $, and $ (20x + 45)^\circ $. Lines BCD and AFE are straight lines, and we need to show that BCD is parallel to AFE. We also need to find the numerical value of the angle $ (20x + 45)^\circ $.\n\n2. **Understanding the problem:** Since BCD and AFE are straight lines, the angles around point C and point F relate to each other. To show that BCD is parallel to AFE, we can use the property that alternate interior angles or corresponding angles are equal when two lines are parallel.\n\n3. **Using the straight line property:** The sum of angles on a straight line is $180^\circ$. So, for line BCD, the angles $ (4x + 15)^\circ $ and $ (30x - 5)^\circ $ add up to $180^\circ$.\n\n$$ (4x + 15) + (30x - 5) = 180 $$\n\n4. **Simplify the equation:**\n$$ 4x + 15 + 30x - 5 = 180 $$\n$$ 34x + 10 = 180 $$\n\n5. **Solve for $x$:**\n$$ 34x = 180 - 10 $$\n$$ 34x = 170 $$\n$$ x = \frac{170}{34} $$\n$$ x = 5 $$\n\n6. **Find the numerical value of $ (20x + 45) $:**\n$$ 20x + 45 = 20(5) + 45 = 100 + 45 = 145 $$\n\n7. **Conclusion:** The angle $ (20x + 45)^\circ $ is $145^\circ$. This confirms the relationship between the angles and supports that BCD is parallel to AFE by the properties of corresponding or alternate interior angles.