I have a square and a pentagon with sides of equal length. These two surfaces touch at one corner (at the origin), forming acute angle and obtuse angles of unknown dimensions between the adjoining sides. I am able to rotate each of these surfaces down (or up) from the horizontal plane along the limbs of the obtuse angle. However I would like to know how to rotate them PRECISELY so that the end points of the acute angle meet. Any help would be appreciated. Thanks.
This is one of the shortcomings of the Rotate tool. It has a precision limit of 0.1 degrees. It would be nice if SketchUp had a construction circle or arc tool, using a true non-segmented circle or arc that would enable us to use inference for the task.
Anssi
Thanks Anssi. I was hoping there was a work-around or some creative geometric solution. Been struggling with this puzzle for months and have tried everything I can think of to resolve it. Obviously unsuccessfully. 
Here’s a workaround with two highly segmented arcs…
You can have a look at these older posts at sketchucation with a lot of discussions in this direction, but I don’t remember if those solutions would work here…
http://sketchucation.com/forums/viewtopic.php?f=15&t=44972&start=300#p403103
http://sketchucation.com/forums/viewtopic.php?f=323&t=17948&p=404020#p403999
http://sketchucation.com/forums/viewtopic.php?f=15&t=53084#p482298
Actually, the precision limit of the protractor is whatever you set in the Model Info Units panel. You can set 0.001 degree if you wish. And you can actually type whatever angle you wish into the VCB for the rotate tool. However, if you are looking for an exact match, even that may not be enough.