Hi,
So I am slightly confused by some of the lettering, but perhaps I am missing something. If you are told that \angle BON = 65, then do you understand the working to find \angle AOC = 30 degrees (I am assuming point C is where point N is)?
If we can accept that, then I think the easiest way to get \angle BOA is just to consider \angle BON = \angle BOA + \angle AON. Solving this yields the same answer of 35 degrees.
I believe they have labeled the 115 degrees by using the fact that line segments BA and ON are parallel and \angle OBA and \angle BON are therefore supplementary ('internal' angles of parallel lines add up to 180 degrees), so they could get 115 from \angle OBA = 180 - 115
Hope that answers your question. If not, let me know and I will respond appropriately.
Kind regards