Help me with GEOMETRY proof please?

DH = DG → triangle DGH is isosceles → <DGH = <DHG

Then by vertical angles we have <BHC = <AGE since those are equal to <DGH and <DHG.

CH = CG - DH

EG = EH - DH

and CG = EH (given), then CH = EG

<3 = <4 (given), then

by the rule of ASA, triangles AEG and BCH are congruent.

So they have the same height, which means AB || CE.

Then AG = BH, and ABGH is an isosceles trapezoid,

and <6 or <GAB = <HBA

and since <5 is part of <HBA, it is smaller than <HBA

and therefore <5 less than <6.

QED

Extra comment:

For the last part, instead of isosceles trapezoid ABGH,

you can substitute isosceles triangle ABD, and use its angles instead.