in the 1st proof
line 2, they GAVE YOU CD || GH so you just say Given
line 3, ABD is a right angle because CD and AE are perpendicular, so your reason is likely def of perpendicular (I'd have to see your book, your list of postulates and theorems to be sure)
line 5, there's no transitive anything here. transitive is a = b, b = c, so a = c (or congruent). the alternate exterior angles are congruent because they're formed by 2 parallel lines cut by a transversal.
line 7, GH and AF are lines (segments?), not angles.
•you have the idea, but how you implement it depends a great deal on what postulates and theorems you have available at this point and what level of detail your teacher expects.
•you're given <AGF is 90°, so you know <DGF is also 90°, but how do you know. do you have a theorem that says all the angles formed by 2 perpendicular lines are right angles, or do you use definition of perpendicular.
•<BGE is a straight angle, measure 180°, but that's not definition of a line. "line" in fact is not defined.
•best way to go is probably say <EGF and <EGD are complementary,
so their sum is 90°, so m<EGD is 30°, then <EGD and <AGB are vertical angles, hence congruent, and so m<AGB is 30.
so much of this depends on the teacher's style that it's hard to give precise advice.