Suppose one angle is 60 degrees as shown in the diagram.
Suppose the angle $s$ is $60$ degrees, then it is easy to calculate
the length of the diagonal and from that to
calculatethe opposite angle in the diagram.
You might like to check your answer by drawing the quadrilateral
accurately, using ruler and compasses only, and then measuring the
angles.
Calculate the other angles of the quadrilateral.
Now calculate the angles of the cyclic quadrilateral formed by
keeping the lengths of the sides the same and changing the angles
so that opposite angles add up to 180 degrees. You might wish to
use a spreadsheet.

The question states that the quadrilateral is convex; this
means that the angles $s$ and $q$ are at most $180$ degrees.
Imagine moving the rods to make the angle $s$ as large or as
small as possible. Find the largest and smallest values of $s$ and
$q$.

To calculate the angles of the cyclic quadrilateral formed by
keeping the lengths of the sides the same and changing the angles
so that opposite angles add up to $180$ degrees you simply need to
use the fact that, in this case, $\cos s =  \cos q$.