The size of the angle x for the triangle ∆DFH on the tangent line EG is equal to 77 degrees
The angle between a chord and a tangent is equal to the angle in the alternate segment, this is the angle subtended by the chord in the opposite side of the previous angle. This angle is also equal to the measure of the intercepted arc divided by 2.
This implies that m∠DFE is equal to m∠DHF, so applying the sum of interior angles of a triangle we can solve for x as follows:
29 + 74 + x = 180
103 + x = 180
x = 180 - 103 {collect like terms}
x = 77°
Therefore, the measure of the angle x in triangle ∆DFH on the tangent line EG is equal to 77 degrees.