You haven't given any definition of what T and N are in relation to the trapezoid (nor really identified which points are which on the trapezoid, though we can infer some of this from step c). So we can't provide a justification for most of these steps.
(c) is presumably because they're the parallel sides of the trapezoid. For (d) I can only think that T, R and I are on the same line, in which case the angles TRG and TIA are congruent because RG and IA are parallel, so they are corresponding angles. For (h), I assume by vertical you mean opposite (it's also called vertically opposite). Vertical angles are always congruent. You just said they're vertical, so this follows.
For (i), from (f) and (h) we have that corresponding angles are congruent, so the triangles must be similar. From (a) we have that GR = GA, so the triangles are congruent by AAS.
For (j), we could infer this from (i). However step (i) isn't strictly necessary; we can derive (j) directly from (f) and (h) - indeed this was the first part of the proof of (i).
