Files from the polygons I would like to intersect are attached.
If you have the lines Intersect in ArcTohttp://desktop.arcgis.com/en/arcmap/latest/tools/analysis-toolbox/intersect.htmolbox will produce a point where two lines intersect or you can split lines with another line, then Split lines several ways
http://desktop.arcgis.com/en/arcmap/latest/tools/data-management-toolbox/split-line-at-point.htm
http://desktop.arcgis.com/en/arcmap/latest/manage-data/editing-existing-features/splitting-lines-manually.htm
loads in fact
http://desktop.arcgis.com/en/arcmap/latest/manage-data/editing-fundamentals/ways-to-split-a-line-feature.htm
Then it is simply a matter of calculating the distance of the split lines, or you could calculate the geodesic distance from the start points to the intersection points.
Loads of options, if your data are in the same coordinate system.
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