Intense Angle Chasing!

We will use complex numbers. Denote the complex number for each point by its lower case character and let the circumcenter be the origin and the circumcircle be the unit circle.

Let the real axis be the line through the circumcenter that is perpendicular to all three parallel lines. This implies that $a = \overline{d} , b = \overline{e} , c = \overline{f}$

This implies that $d = \frac{1}{a} , e = \frac{1}{b} , f = \frac{1}{c}$

It is well-known that if $a, b$ are two complex numbers on the unit circle, then the reflection of complex number $z$ over line $AB$ is $a+b-ab\overline{z}$ .

This implies that

$s = a+b - abc$

$t = a + c - abc$

$u = a + b - abc$

It is well-known that if the circumcenter is the origin, then

$h = a + b + c$ .

Remark that these four points are concyclic by noticing their symmetry. It is easy to see that their center is $a + b + c - abc$ . You can confirm this by subtracting the center with each point and taking the magnitude, which in this case, is one.

Therefore, since the two angles in cyclic quadrilateral $SHUT$ are not supplementary, the other two angles must be supplementary to the given angles. Therefore the other two angles are $141$ degrees and $89$ degrees implying a positive difference of $\boxed{52}$

$\textit{This is an altered version of a problem from MOP }$