1. **State the problem:** We have two parallel lines PQ and ST cut by a transversal. Given angles are $35^\circ$ at PQ and $140^\circ$ at ST. We need to find the value of $x$, the alternate interior angle to $140^\circ$ at PQ. 2. **Recall angle rules:** When two parallel lines are cut by a transversal, alternate interior angles are equal. 3. **Identify alternate interior angles:** The angle $x$ at PQ is alternate interior to the $140^\circ$ angle at ST. 4. **Apply the rule:** Since alternate interior angles are equal, $$x = 140^\circ$$ 5. **Check for consistency:** The $35^\circ$ angle at PQ is adjacent to $x$ on a straight line, so their sum should be $180^\circ$. Calculate: $$35^\circ + x = 180^\circ$$ $$x = 180^\circ - 35^\circ = 145^\circ$$ This contradicts the previous step, so the $140^\circ$ angle at ST is not alternate interior to $x$ but rather the supplementary angle to the alternate interior angle. 6. **Correct interpretation:** The angle adjacent to $x$ at PQ is alternate interior to $140^\circ$ at ST, so $$x + 35^\circ = 140^\circ$$ 7. **Solve for $x$:** $$x = 140^\circ - 35^\circ = 105^\circ$$ **Final answer:** $$x = 105^\circ$$