I need to add this Lake County townships map into a shapefile that contains other counties' townships. The shapefile's fields are: Township, Shape, Shape_Length, Shape_Area and County.
How can I do this in ArcMap 10.4?
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DPSSpatial's comment that this data might already exist in vector form is worth investigating as that's the easiest option. I would start by looking at the US Census.
If not, you'll have to digitize (trace) the features you want from your TIFF to create a vector shapefile with the same fields before combining it with your other data.