1. **State the problem:** We need to find the value of angle $y$ at point $G$ inside the circle, given angles $72^\circ$ and $108^\circ$ at point $F$, and that line $EFG$ is parallel to segment $JH$. 2. **Identify key facts and formulas:** - Since $EFG$ is parallel to $JH$, corresponding angles formed by a transversal are equal. - The sum of angles on a straight line is $180^\circ$. - The sum of angles in a triangle is $180^\circ$. 3. **Analyze angles at point $F$:** - Angle $108^\circ$ is outside the circle at $F$ on line $FE$. - Angle $72^\circ$ is inside the circle at $F$. - Since $EFG$ is a straight line, angles $108^\circ$ and $72^\circ$ add to $180^\circ$, confirming the straight line. 4. **Use parallel lines property:** - Because $EFG \parallel JH$, angle $108^\circ$ at $F$ corresponds to angle $JHG$ at $H$, so angle $JHG = 108^\circ$. 5. **Consider quadrilateral $GHJF$ inscribed in the circle:** - Opposite angles of a cyclic quadrilateral sum to $180^\circ$. - Angles at $H$ and $F$ are opposite angles. - Angle at $F$ inside the circle is $72^\circ$, so angle at $H$ inside the circle is $180^\circ - 72^\circ = 108^\circ$. This matches the previous step, confirming consistency. 6. **Find angle $y$ at $G$:** - Triangle $GHF$ has angles at $H = 108^\circ$, at $F = 72^\circ$, and at $G = y$. - Sum of angles in triangle $GHF$ is $180^\circ$, so $$y + 108^\circ + 72^\circ = 180^\circ$$ $$y + 180^\circ = 180^\circ$$ $$y = 0^\circ$$ This suggests $y$ is $0^\circ$, which is not possible for an angle inside a triangle. We must reconsider the triangle. 7. **Re-examine the triangle:** - The angle $y$ is at $G$, inside the circle. - The angle at $H$ is $108^\circ$. - The angle at $F$ inside the circle is $72^\circ$. - The triangle formed is $GHF$. - Sum of angles: $$y + 108^\circ + 72^\circ = 180^\circ$$ $$y = 180^\circ - 180^\circ = 0^\circ$$ This contradiction means $y$ is not an interior angle of triangle $GHF$. Instead, $y$ is the angle subtended by chord $JF$ at point $G$. 8. **Use the property of angles subtended by the same chord:** - Angle $y$ at $G$ and angle $72^\circ$ at $F$ subtend the same chord $JH$. - Angles subtended by the same chord in the same segment are equal. - Therefore, $$y = 72^\circ$$ **Final answer:** $$\boxed{72^\circ}$$