All of what the others have said is true, but they left something out. You can still use an Ancestry DNA Match for Triangulation IF you download the raw DNA files in question and upload them to GEDmatch. Once uploaded to GEDmatch, they can be "Triangulated" with others who have also uploaded to GEDmatch. I see that your DNA is already there, so it's just a matter of running some of the tools to see if you can match anyone. Since your mother is private, you will need to check this yourself.
As was stated by Matthew, you do not need triangulation for 3rd cousin or closer relationships so Ancestry will work in those situations.