1. **State the problem:** We need to find the size of angle $x$ in a figure with two parallel horizontal lines and two transversals forming angles $137^\circ$ and $65^\circ$ as shown. 2. **Angle facts used:** - Angles on a straight line sum to $180^\circ$. - Alternate interior angles between parallel lines are equal. - The sum of angles in a triangle is $180^\circ$. 3. **Find the angle adjacent to $137^\circ$ on the bottom line:** $$180^\circ - 137^\circ = 43^\circ$$ 4. **Identify the angle corresponding to $65^\circ$ below the bottom line:** This angle is an exterior angle to the triangle formed by the transversals and the parallel lines. 5. **Calculate the interior angle adjacent to $65^\circ$ on the bottom line:** $$180^\circ - 65^\circ = 115^\circ$$ 6. **Use alternate interior angles:** The angle on the top line corresponding to $115^\circ$ is also $115^\circ$ because the lines are parallel. 7. **Sum of angles in the triangle formed at the top vertex:** $$x + 43^\circ + 115^\circ = 180^\circ$$ 8. **Solve for $x$:** $$x = 180^\circ - 43^\circ - 115^\circ$$ $$x = 22^\circ$$ **Final answer:** $$\boxed{22^\circ}$$