1. **State the problem:** We have two parallel lines $a$ and $b$ cut by two intersecting lines, creating angles labeled $123^\circ$, $81^\circ$, $116^\circ$, $x^\circ$, and $y^\circ$. We need to find the value of $x + y$. 2. **Identify relationships:** Since lines $a$ and $b$ are parallel, corresponding and alternate interior angles are equal. 3. **Use given angles:** The angle $123^\circ$ is given, and the handwritten annotation shows $57^\circ$ near it, which is the supplementary angle because $180^\circ - 123^\circ = 57^\circ$. 4. **Find $x$:** The angle $x^\circ$ is vertically opposite to the angle labeled $116^\circ$, so $x = 116^\circ$. 5. **Find $y$:** The angle $y^\circ$ is supplementary to the $81^\circ$ angle because they form a linear pair, so $y = 180^\circ - 81^\circ = 99^\circ$. 6. **Calculate $x + y$:** $$x + y = 116^\circ + 99^\circ = 215^\circ$$ **Final answer:** $x + y = 215^\circ$.