1. **State the problem:** We are given two parallel lines $a$ and $b$ cut by a transversal, with angles $ (2x - 5)^\circ $ on line $a$ and $ (x + 20)^\circ $ on line $b$. We need to find the value of $x$. 2. **Identify the relationship:** Since lines $a$ and $b$ are parallel, the angles given are corresponding angles and therefore equal. 3. **Set up the equation:** $$ 2x - 5 = x + 20 $$ 4. **Solve for $x$:** \begin{align*} 2x - 5 &= x + 20 \\ 2x - \cancel{x} - 5 &= \cancel{x} + 20 \\ x - 5 &= 20 \\ x - \cancel{5} &= 20 + \cancel{5} \\ x &= 25 \end{align*} 5. **Check the answer:** Substitute $x=25$ back into the angles: \begin{align*} 2x - 5 &= 2(25) - 5 = 50 - 5 = 45^\circ \\ x + 20 &= 25 + 20 = 45^\circ \end{align*} Both angles are equal, confirming the solution. **Final answer:** $x = 25$