To solve this problem you will need to use the sine and cosine rules. Four rods are hinged at their ends to form a convex quadrilateral with sides of length 3, 4, 5 and 6. 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

Suppose the angle

s

is 60 degrees, then it is easy to calculate the length of the diagonal and from it all the angles 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.

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, $\cos s = - \cos q$ .