Explanation:
to point out : the small triangle inside the circle is an equilateral triangle (all 3 sides are equally long). that means all 3 angles are equally large.
as the sum of all angles in a triangle is always 180°, and we have 3 equally large angles, that means each angle is
180/3 = 60°.
the left side of that triangle is the radius connection from the center of the circle to a tangent on the circle, so the angle between these 2 lines (radius and tangent) is 90°.
that means that the left angle of the outer triangle (with the angle x) is complementary (together they have 80°) to the 60° inner angle of the small triangle :
left outer angle = 90 - 60 = 30°
and we can see that the large angle of the outer triangle is the supplementary angle (together they have 180°) to the right angle of the inner triangle :
large angle = 180 - 60 = 120°
simply because the sum of all angles around a single point on one side of a line is 180° (because that line can be seen as the diameter of a circle, one side of the line represents then a half-circle, and every half-circle has 180°).
so, we know the left angle of the triangle with angle x : 30°.
we know the large angle of that triangle : 120°.
again the sum of all angles in a triangle is always 180°.
so,
180 = 30 + 120 + x
x = 180 - 30 - 120 = 30°
FYI
since both angles at the baseline are equally large (30°), this makes this an isoceles triangle (both legs are equally long).
so, the length of the line from the circle arc to the vertex of x (the right leg of the outer triangle) is equal to the length of the secant between both intersections with the circle arc (the upper side of the inner triangle = the bottom side of the outer triangle).