Explanation:
8.
the angle at a vertex (crossing point) outside of the circle (when both lines intersect with the circle arc, which is also the case for tangents) is half of the difference of both arc angles (the outer and the inner angles).
the outer angle is 250°. since the full circle is 360°, this makes the inner arc angle 360-250 = 110°.
the difference is 250-110 = 140.
and half of that difference is
140/2 = 70°.
so, x = 70°
9.
see 8.
the angle at the vertex outside the circle is half of the difference between outer and inner arc angle.
in fact, this is again about tangents (despite the picture), because the outer angle + the inner angle is again 360° (the full circle). this can only be, if the 2 lines are tangents splitting the outer and inner arc angles into 2 parts of 360° without remainder.
320 - 40 = 280
280/2 = 140
x = 140°
10.
see 8. and 9.
it is an extreme example, where the vertex of the 2 intersecting lines is exactly on the circle arc and the tangent point of the second line.
this makes the inner arc angle = 0°.
remember, the vertex angle (55°) is half of the difference of the outer (x) and inner (0°) arc angle.
55 = (x - 0)/2
110° = x
11.
remember the table question, where we found that the circle internal angle of 2 intersecting lines is half of the sum of both intersected arc angles ?
this is exactly this case again.
(120 + 50)/2 = 170/2 = 85°
x = 85°.
12.
first, like in 8. and 9., we determine the vertex angle. x is then the supplementary angle (together they have 180°) to this vertex angle, because the sum of all angles around one point on one side of a line is always 180°.
the vertex angle is
(125 - 15)/2 = 55°
x + 55 = 180
x = 125°.
13.
see 11.
the circle internal vertex angle is half of the sum of both circle arcs.
x = (70 + 30)/2 = 100/2 = 50°.
14.
again, an extreme case, where the vertex is on the circle arc. so, the second arc angle is 0°.
see also 12.
x is the supplementary angle of the vertex angle.
the vertex angle is
(80 + 0)/2 = 80/2 = 40°.
x + 40 = 180
x = 140°.
15.
it is difficult to read. I see the outer arc circle as 140°.
see 8., 9., 10.
the vertex angle x is
x = (140 - 70)/2 = 70/2 = 35°
do I sound like a broken record ? I at least feel that way. you should be able to apply the explanations from the previous questions to solve the new ones yourself.
what is not clear yet ? where do you need further explanations ?