1. **State the problem:** We need to find the size of angle $x$ in a figure with intersecting lines and given angles $65^\circ$ and $137^\circ$. 2. **Identify angle facts used:** - Angles on a straight line sum to $180^\circ$. - Alternate angles are equal. - Exterior angle of a triangle equals the sum of the two opposite interior angles. 3. **Use the straight line fact for the $137^\circ$ angle:** $$\text{Adjacent angle} = 180^\circ - 137^\circ = 43^\circ$$ 4. **Use alternate angles fact:** The $65^\circ$ angle is alternate to an angle adjacent to $x$, so the angle adjacent to $x$ is also $65^\circ$. 5. **Sum of angles around point or in triangle:** Since $x$ and the adjacent $65^\circ$ angle form a straight line, $$x + 65^\circ = 180^\circ$$ 6. **Solve for $x$:** $$x = 180^\circ - 65^\circ = 115^\circ$$ **Final answer:** $$x = 115^\circ$$