1. **Problem Statement:** We need to find the measure of angle $\angle GMK$ given two expressions for angles related by parallel lines and a transversal. 2. **Given:** - $\angle HLF = (22x + 4)^\circ$ - $\angle JMK = (96 - 2x)^\circ$ 3. **Important Rule:** When two parallel lines are cut by a transversal, alternate interior angles are equal. 4. **Identify Angles:** - $\angle HLF$ and $\angle JMK$ are alternate interior angles because lines HL and JK are parallel and FLM is the transversal. 5. **Set up equation:** $$22x + 4 = 96 - 2x$$ 6. **Solve for $x$:** $$22x + 2x = 96 - 4$$ $$24x = 92$$ $$x = \frac{92}{24} = \frac{23}{6} \approx 3.833$$ 7. **Find $\angle GMK$:** - $\angle GMK$ is adjacent to $\angle JMK$ on a straight line, so they are supplementary. - Supplementary angles sum to $180^\circ$. 8. **Calculate $\angle JMK$ value:** $$96 - 2x = 96 - 2 \times \frac{23}{6} = 96 - \frac{46}{6} = 96 - 7.6667 = 88.3333^\circ$$ 9. **Calculate $\angle GMK$:** $$\angle GMK = 180^\circ - 88.3333^\circ = 91.6667^\circ$$ **Final answer:** $$m\angle GMK \approx 91.67^\circ$$