1. **State the problem:** We are given two parallel lines cut by a transversal, forming a triangle with angles labeled $x + 25^\circ$ and $x - 5^\circ$. We need to find the value of $x$. 2. **Recall the rule:** When two parallel lines are cut by a transversal, alternate interior angles are equal, and the sum of angles in a triangle is $180^\circ$. 3. **Set up the equation:** The triangle formed has three angles: $x + 25^\circ$, $x - 5^\circ$, and the third angle formed by the transversal. Since the two given angles are adjacent to the parallel lines, the third angle is the exterior angle to one of these, so the sum of the three angles is $180^\circ$. 4. **Write the sum of angles:** $$ (x + 25) + (x - 5) + \text{third angle} = 180 $$ 5. **Find the third angle:** The third angle is the angle between the transversal and the segment connecting the two intersection points. Since the two given angles are on the same side of the transversal and the lines are parallel, the third angle equals the difference between the two given angles: $$ \text{third angle} = (x + 25) - (x - 5) = x + 25 - x + 5 = 30 $$ 6. **Substitute the third angle:** $$ (x + 25) + (x - 5) + 30 = 180 $$ 7. **Simplify:** $$ x + 25 + x - 5 + 30 = 180 $$ $$ 2x + 50 = 180 $$ 8. **Solve for $x$:** $$ 2x = 180 - 50 $$ $$ 2x = 130 $$ $$ x = \frac{130}{2} $$ $$ x = 65 $$ **Final answer:** $x = 65$