Mid way through a project when it is noticed two lines were not divided in CAD. Instead of going back into CAD, exporting, ungrouping, copy pasting, snapping, drag select undo deleting;

How does Inkscape break the magenta lines over the blue crescent?

 Thank you kindly

You need one of Inkscape's boolean operations. Select two paths then [Path > Cut Path]. The upper path is used as a scalpel to cut the lower path where they intersect. The scalpel is discarded after use so you'll need to make your own before each cut. It's just good hygiene.

Thanks @Paddy_CAD. I was able to break the lines from what you told me.  For some reason the magenta arc, once cut, had to be flipped vertically and horizontally then snapped back onto path. Would there be a way to avoid this flipping from happening?

That's very strange. When I edited your drawing, both arc fragments remained in their original positions. I've no idea why the pieces would be inverted like that.

I noticed the flipping happens to one half of cut arcs but not cut lines; I guess because there is no difference when a straight line is flipped vertical and horizontal. The cut path tool works though. Thank you kindly @Paddy_CAD