What's the relationship between angle $x+y$ and $w+v$ in this picture given two triangles with one side parallel formed above and below a common line?

Question

Is

I. $x+y> w+v$

II. $x+y< w+v$

III. $x+y= w+v$

IV. cannot be determined?

Why is $x$ equal to $v$?

My idea was that the x+y is not necessarily w+v, because if you move the two triangles closer so the third vertex of the triangle with vertices A and B shares a vertex with the triangle with vertices C and D, the path from B to C is not necessarily a straight line.

Answer 1

If the first two inequalities are true, then the third equality automatically holds because $x+y-v-w$ cannot be positive and negative at the same time, needs to vanish.

You have not stated whether length and curvature can change.

If I understood correctly, because the angle sum is constant each term representing angle need not be equal to the corresponding angle and there are many other possibilities... like we can assume length of a straight line AB changing and moving to position ab, as also CD moving to position cd... they can rotate arbitrarily.