In ArcGIS use the Feature Vertices To Points tool to create two different datasets, an end vertices dataset and a start vertices dataset. Select all those start vertices that intersect lakes. Select all end vertices that intersect lakes. Select all line that intersect the selected start vertices. From those selected lines select all lines that intersect the selected end vertices. Now you should have a line selection that includes any lines connecting lakes, and all lines inside of lakes. Remove from the selection any lines contained within the lake polygons.

You mention your data is dissolved. Are the line segmenst between the lakes the same record as the line segments before or after a lake? If so, some geometry editing will be required. Maybe a screen shot of your rivers symbolized by ObjectID would be useful.

In ArcGIS use the Feature Vertices To Points tool to create two different datasets, an end vertices dataset and a start vertices dataset. Select all those start vertices that intersect lakes. Select all end vertices that intersect lakes. Select all line that intersect the selected start vertices. From those selected lines select all lines that intersect the selected end vertices. Now you should have a line selection that includes any lines connecting lakes, and all lines inside of lakes. Remove from the selection any lines contained within the lake polygons.

You mention your data is dissolved. Are the line segments between the lakes the same record as the line segments before or after a lake? If so, some geometry editing will be required. Maybe a screen shot of your rivers symbolized by ObjectID would be useful.

Use the select by location features in ESRI ArcGIS. Select by location has an optioin called Contain. With Contain your line must be completely contained by the polygon but they can share geometries at the border.

In ArcGIS use the Feature Vertices To Points tool to create two different datasets, an end vertices dataset and a start vertices dataset. Select all those start vertices that intersect lakes. Select all end vertices that intersect lakes. Select all line that intersect the selected start vertices. From those selected lines select all lines that intersect the selected end vertices. Now you should have a line selection that includes any lines connecting lakes, and all lines inside of lakes. Remove from the selection any lines contained within the lake polygons.

You mention your data is dissolved. Are the line segmenst between the lakes the same record as the line segments before or after a lake? If so, some geometry editing will be required. Maybe a screen shot of your rivers symbolized by ObjectID would be useful.