Flight Instructor with student reported a near miss on a training flight in VMC weather during climb out from ELN non-towered airport. The instructor performed an evasive maneuver to provide separation when the other aircraft did not turn away from the instructor pilot aircraft.

Narrative

I am flying with a student practicing the ODP out of Ellensburg. We made an initial call that we will be turning toward the VOR. Once we got to the VOR; still climbing; we made a call that we will be holding West. I visually clear right and see Aircraft Y flying parallel to us and 500 ft low a few miles away. I deem it safe to turn and have my student turn. Right after we made the call and turned Aircraft Y gets on the mic 'turning sharp left turn'. I continue the turn to the right and Aircraft Y continues straight at us. Aircraft Y then makes a call 'Aircraft X do you have Aircraft Y in sight or on the ADS-B'. I had them on ADS-B but; no longer could see them. I answer back 'we have you on ADS-B but can't see you; we will continue our turn to the right.' Pilot then gets aggressive on the mic and says 'you should not have done that'. I asked 'why did you turn left?'. Pilot responds with 'BECAUSE I WAS TURNING ON COURSE AND MADE A CALL'.A lot of changes could have been made to help with this encounter; but overall; we don't need aggressive flyers. Good lesson for my student though!