Final answer:
To determine angle RGO, geometric properties of congruent triangles and the relationship between arc length and rotation angle in a circle are used.
Step-by-step explanation:
To find the measure of angle RGO for circle H, we need to employ our understanding of geometry and trigonometry. Given the information about congruent triangles, similar proportions, and the relationship between arc length, radius, and rotation angle, we can calculate the desired angle. Since HKD and KFD are congruent triangles, and all four triangles mentioned (HKD, KFD, GFC, and AHD) are congruent, we can infer that the arcs they subtend have equal angles when at the center of circle H.