Central angle AOH is made up of the 3 red central angles.
Each of those three central angles measures 360°÷10 or 36°.

So the 3 of them have measure 3×36° or 108°, so angle AOH
has measure 108°.  Therefore minor arc AH which angle ADH 
subtends has measure 108°.  

Angle ADH is an inscribed angle and an inscribed angle gas 
the measure of one-half of the measure of its subtended arc.
Angle ADH subtends the same arc which central angle AOH 
subtends. Thus the measure of angle ADH is one-half of 108° 
and is therefore 54°. 

Edwin