1. **Problem statement:** We have a circle with center $P$ and diameters $FH$ and $EG$. We need to find the measure of arc $DGE$. 2. **Given information:** - $FH$ and $EG$ are diameters, so they pass through the center $P$ and their endpoints lie on the circle. - The central angle between $PH$ and $PG$ is $38^\circ$. - There is a right angle ($90^\circ$) between $PD$ and $PH$. 3. **Key concepts:** - The measure of an arc is equal to the measure of its central angle. - Since $FH$ and $EG$ are diameters, points $F$, $H$, $E$, and $G$ lie on the circle and $P$ is the center. - The central angle between two points on the circle is the angle formed at the center by the radii to those points. 4. **Step-by-step solution:** - Since $FH$ is a diameter, points $F$ and $H$ are opposite ends of the circle. - Similarly, $EG$ is a diameter, so points $E$ and $G$ are opposite ends. - The central angle between $PH$ and $PG$ is $38^\circ$, so the angle $\angle HPG = 38^\circ$. - The right angle between $PD$ and $PH$ means $\angle DPH = 90^\circ$. 5. **Find the central angle $\angle DPG$:** - $\angle DPG = \angle DPH + \angle HPG = 90^\circ + 38^\circ = 128^\circ$. 6. **Arc $DGE$ corresponds to central angle $\angle DPG$ because $E$ and $G$ are endpoints of diameter $EG$ and $D$ lies on the circle between $G$ and $E$. So the measure of arc $DGE$ is $128^\circ$. **Final answer:** $$\boxed{128^\circ}$$ The correct choice is C.