1. **State the problem:** We are given a circle H with angles $m\angle DHG = 11x - 36$ and $m\angle GHF = 3x + 12$. We need to find the value of $x$ and the measure of $\angle DGH$. 2. **Understand the relationship:** Since $\angle DHG$ and $\angle GHF$ are angles around point H on the circle, and assuming they are adjacent angles on a straight line or part of a triangle, their sum might be 180° or 360° depending on the figure. Usually, if these two angles are adjacent and form a straight line, their sum is 180°. 3. **Set up the equation:** Assuming $\angle DHG + \angle GHF = 180^\circ$, we write: $$11x - 36 + 3x + 12 = 180$$ 4. **Simplify the equation:** $$14x - 24 = 180$$ 5. **Isolate $x$:** $$14x = 180 + 24$$ $$14x = 204$$ 6. **Divide both sides by 14:** $$x = \frac{204}{14}$$ $$x = \frac{\cancel{204}^{17} \times 12}{\cancel{14}^1 \times 7} = \frac{102}{7} \approx 14.57$$ 7. **Find the measure of $\angle DHG$:** $$m\angle DHG = 11x - 36 = 11 \times \frac{102}{7} - 36 = \frac{1122}{7} - 36 = 160.29 - 36 = 124.29^\circ$$ 8. **Find the measure of $\angle GHF$:** $$m\angle GHF = 3x + 12 = 3 \times \frac{102}{7} + 12 = \frac{306}{7} + 12 = 43.71 + 12 = 55.71^\circ$$ 9. **Find the measure of $\angle DGH$:** If $\angle DGH$ is the third angle in triangle DHG, then: $$m\angle DGH = 180 - m\angle DHG - m\angle GHF = 180 - 124.29 - 55.71 = 0^\circ$$ This suggests $\angle DGH$ is 0°, which is unlikely, so $\angle DHG$ and $\angle GHF$ might be vertical angles and equal. 10. **Alternative assumption:** If $m\angle DHG = m\angle GHF$, then: $$11x - 36 = 3x + 12$$ $$11x - 3x = 12 + 36$$ $$8x = 48$$ $$x = 6$$ 11. **Calculate angles with $x=6$:** $$m\angle DHG = 11(6) - 36 = 66 - 36 = 30^\circ$$ $$m\angle GHF = 3(6) + 12 = 18 + 12 = 30^\circ$$ 12. **Find $\angle DGH$:** If triangle DHG is formed, sum of angles is 180°: $$m\angle DGH = 180 - 30 - 30 = 120^\circ$$ **Final answers:** $$x = 6$$ $$m\angle DGH = 120^\circ$$