If you are talking about polylines, you can try the Miminum Bounding Geometry tool with the RECTANGLE_BY_WIDTH option and check the checkbox "Add geometry characteristics as attributes to output". You may not need the output geometry, but the field MBG_Orientation might be what you want? You can join the output with the polyline input through the ORIG_FID field.You can find the Miminum Bounding Geometry tool in Data Management toolbox - Features toolset.
Does anyone know of a way to get the bearings of a parcel to be displayed on the map? or a way to access the bearing info? I.e. I'm trying to find the bearing of a property line...
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