If you know the point on the circle that is colliding with the other circle or the line, you can determine the angle between whatever direction the circle is "facing" (if the circle, and thus the individual arcs, can rotate around the center) and the collision point. The arcs can then just be thought of as angle-ranges (0 through PI/2, Pi/2 through 3Pi/2, etc.).

If you got the local closest point on the circle, just use arctan2 to get the angle for that point.

Secondly bring that angle into range 0 - 2*PI (google: normalize angle radians)

Detect region/arc from that normalized angle. Done.

Do you have the point of intersection worked out?

atan2 can give you the angle if you feed it the y,x position relative to the center of the circle. The angle it gives will consider zero degrees/radians to be pointing toward the right.

Each one of your arcs has a start angle and an end angle, so just look to see which range your angle is in.