Explanation:
if we extend the line VT to reach also the opposite side of the circle, we see that this line cuts the circle in half.
the whole arc angle of a half-circle is 360/2 = 180°.
now we can use the exterior angle rule :
the vertex angle (V) = 1/2 × difference of the angles of the intersected arcs.
the intersected arcs are
shorter = angle at T : the arc from U to the circle intersection with the line VT.
longer : the arc from U to the circle intersection with the extended line VT.
shorter + longer = 180°
longer = 180 - shorter
30 = 1/2 × ((180 - shorter) - shorter) =
= 1/2 × (180 - 2×shorter) = 90 - shorter
-60 = -shorter
shorter = 60°
the angle at T = 60°.