Nice title :).
So, we know that JI and JX are perpendicular (first statement). That just means that they mean at an angle of 90*.
That means CIJ has an angle of 150*-90*=60*. Therefore, as IG bisects CIJ , both CIG and GIJ have angle 60*/2=30*.
Half a circle has 180*. GIR is just short of CIG to be a half circle.
Therefore
180*-mCIG=mGIR
We know mCIG was 30*, so
150*=mGIR