Altitudes!

Since  angle(BAC) = 54^o  and  angle(ABC) = 52^o,  angle(BCA)  =  180^o - 54^o - 52^o  =  74^o.

Look at quadrilateral(DCEH):

  --  since AD is an altitude, angle(CDA) = 90^o

  --  since BE is an altitude, angle(CEB) = 90^o 

  --  since there are 360^o in a quadrilateral:  angle(CDA) + angle(CEB) + angle(BCA) + angle(DHE)  =  360^o

       --  substituting, we get angle(DHE) = 106^o

angle(AHB) and angle(DHE) are vertical angles   --->   angle(DHE)  =  106^o

∠AHB = ∠EHD

∠DCE = 180° - 54° - 52° = 74°

To find the angle EHD, we can use the interior angles of a quadrilateral EHDC ( 360° )

∠EHD = 360° - ( 90° + 74° + 90° )