Given : AG and IC are common internal tangents of H and B . HI =8 , ED =2 and EF = ED . What is the measure of AE?

The figure is completely symmetrical about the point E. Thus, AE = IE. Of course, angle I is a right angle, so we have (by the Pythagorean theorem)
  HI^2 + IE^2 = HE^2
  8^2 + IE^2 = 10^2
  IE^2 = 100 - 64 = 36
  IE = 6

The measure of AE is 6.