1. **Stating the problem:** We need to find the measures of angles $\angle G$, $\angle H$, and $\angle I$ in triangle $GHI$. 2. **Given information:** There is an angle of $110^\circ$ at point $X$ and an angle of $20^\circ$ at point $Y$ formed by two line segments. These points relate to the triangle $GHI$. 3. **Assumption:** Since the problem does not provide explicit angle measures inside triangle $GHI$, but shows external angles $110^\circ$ and $20^\circ$, we can infer these are related to the triangle's interior angles by the exterior angle theorem. 4. **Exterior angle theorem:** The measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles. 5. **Calculate $m\angle G$:** Suppose $110^\circ$ is an exterior angle at vertex $G$, then $$m\angle G = 110^\circ - m\angle H$$ 6. **Calculate $m\angle H$:** Suppose $20^\circ$ is an exterior angle at vertex $H$, then $$m\angle H = 20^\circ - m\angle I$$ 7. **Sum of interior angles:** The sum of interior angles in triangle $GHI$ is $$m\angle G + m\angle H + m\angle I = 180^\circ$$ 8. **Substitute $m\angle G$ and $m\angle H$ from steps 5 and 6 into the sum:** $$ (110^\circ - m\angle H) + m\angle H + m\angle I = 180^\circ $$ 9. **Simplify:** $$ 110^\circ + m\angle I = 180^\circ $$ 10. **Solve for $m\angle I$:** $$ m\angle I = 180^\circ - 110^\circ = 70^\circ $$ 11. **Find $m\angle H$ using step 6:** $$ m\angle H = 20^\circ - m\angle I = 20^\circ - 70^\circ = -50^\circ $$ Since angle measures cannot be negative, this suggests a misinterpretation of the problem or the given angles. **Conclusion:** Without additional information or a diagram, the exact measures of $\angle G$, $\angle H$, and $\angle I$ cannot be determined from the given data. Please provide more details or a diagram for precise calculation.