1. **Problem statement:** Find the angle $x$ in each diagram where two parallel lines are cut by a transversal, and a known angle is given on the opposite parallel line. 2. **Key rule:** When two parallel lines are cut by a transversal, corresponding angles are equal, and alternate interior angles are equal. 3. **Solution for (a):** Given angle = $125^\circ$ on the bottom line, angle $x$ is on the top line on the same side of the transversal. By the corresponding angles rule, $x = 125^\circ$. 4. **Solution for (b):** Given angle = $57^\circ$ on the bottom line, angle $x$ is on the top line but on the opposite side of the transversal. By the alternate interior angles rule, $x = 57^\circ$. 5. **Solution for (c):** Given angle = $70^\circ$ on the bottom line, angle $x$ is on the top line on the same side of the transversal. By the corresponding angles rule, $x = 70^\circ$. 6. **Solution for (d):** Given angle = $105^\circ$ on the bottom line, angle $x$ is on the top line on the opposite side of the transversal. By the alternate interior angles rule, $x = 105^\circ$. 7. **Solution for (e):** Given angle = $53^\circ$ on the bottom line, angle $x$ is on the top line on the same side of the transversal. By the corresponding angles rule, $x = 53^\circ$. 8. **Solution for (f):** Given angle = $133^\circ$ on the bottom line, angle $x$ is on the top line on the opposite side of the transversal. By the alternate interior angles rule, $x = 133^\circ$. **Final answers:** (a) $x = 125^\circ$ (b) $x = 57^\circ$ (c) $x = 70^\circ$ (d) $x = 105^\circ$ (e) $x = 53^\circ$ (f) $x = 133^\circ$