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I have two line datasets. Both the datasets are falling in the same region. Now few of them are matching spatially with each other when I viewed them from top. I wanted to identify all the lines which are matching with each other (both partially and fully). None of them are having common attributes.

How can I do this?

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To find identical overlaps you could try Select Layer By Location with overlap_type = ARE_IDENTICAL_TO:

The features in the input layer will be selected if they are identical (in geometry) to a selecting feature.

To find partial overlaps you can use Intersect:

Computes a geometric intersection of the input features. Features or portions of features which overlap in all layers and/or feature classes will be written to the output feature class.

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Have a look at the OpenJUMP RoadMatcher plugin http://wiki.openstreetmap.org/wiki/RoadMatcher. I suppose it can make what you want but read the PDF manual https://sourceforge.net/p/jump-pilot/code/HEAD/tree/plug-ins/RoadMatcher/trunk/doc/RoadMatcher%20User%20Guide.pdf and consider yourself.

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