I'll go ahead and help you with what I can. :)
ADEG: Rectangle
What we know: all sides are 90 degree angles, opposite sides are parallel (two pairs of opposite parallel sides) by definition of a rectangle. This also means that AG is congruent/parallel to DE, and GE is congruent/parallel to AD.
Segment GE bisects (splits in half) both segments BC and AB (given).
The transitive property: GE bisects segment BC and AB, which means that the resulting two segments from AB, BG and GA, are congruent to each other. Not only this, but BE is congruent to EC. So, if we already know that BG is congruent to GA, and GA is congruent to DE, then, by transitive property of congruence, BG is congruent to DE.
So, one side (hypotenuse, longest side) done. BE is congruent to EC.
Next -- BG is congruent to DE, as we found. Two sides done.
Last but not least, you can see that BEG and DEC are vertical angles. Now you know that these angles are congruent.
Of course, by construction, you can extend AD from point D to the right and find that its exterior angle, EDC, is congruent to GAD by the corresponding angles theorem. You could also use construction again and extend AG by G upwards and find that BGE is congruent to GAD by, again, corresponding angles theorem.
What we know:
Angle BGE is congruent to EDC, both 90 degrees.
BEG and DEC are vertical angles, both congruent.
BG is congruent to DE.
BE is congruent to EC.
GBE is congruent to DEC by corresponding angles theorem.
So you can use ASA (angle B, side BG, and angle G) and (angle E, side ED and angle D) for the rest. Then make sure to use CPCTC (congruent parts of congruent triangles are congruent). :)
Good luck!