1. **State the problem:** We need to find the measure of angle $\angle MON$ given two algebraic expressions for angles formed by intersecting lines. 2. **Identify the angles:** The problem gives two angles: - One angle is $(12x + 30)^\circ$ - The other angle is $(9x + 48)^\circ$ 3. **Understand the relationship:** Since the lines intersect at point $O$, the angles $(12x + 30)^\circ$ and $(9x + 48)^\circ$ are vertical angles or supplementary angles depending on the figure. Given the problem context, these two angles are adjacent and form a straight line, so they are supplementary. 4. **Use the supplementary angle rule:** Supplementary angles add up to $180^\circ$. So, $$ (12x + 30) + (9x + 48) = 180 $$ 5. **Solve for $x$:** $$ 12x + 30 + 9x + 48 = 180 $$ $$ 21x + 78 = 180 $$ $$ 21x = 180 - 78 $$ $$ 21x = 102 $$ $$ x = \frac{102}{21} = \frac{34}{7} \approx 4.857 $$ 6. **Find $m\angle MON$:** The angle $\angle MON$ corresponds to $(9x + 48)^\circ$. Substitute $x = \frac{34}{7}$: $$ m\angle MON = 9\left(\frac{34}{7}\right) + 48 = \frac{306}{7} + 48 = \frac{306}{7} + \frac{336}{7} = \frac{642}{7} \approx 91.71^\circ $$ **Final answer:** $$ m\angle MON \approx 91.71^\circ $$