$PQRS$ is a cyclic quadrilateral. If $\angle P$ is 4 times $\angle R $, and $\angle S $ is 3 times $\angle Q$, then the average of $\angle Q$ and $\angle R $ is:

40.5°

According to question

= \(\angle\)P = 4 x \(\angle\)R

= \(\angle\)P : \(\angle\)R = 4 : 1

By using the property

= \(\angle\)P + \(\angle\)R = 180

= 4x + x = 180

= 5x = 180

= x = 36

= \(\angle\)R = \({36}^\circ\)

Now,

= \(\angle\)S = 3 x \(\angle\)Q

= \(\angle\)S : \(\angle\)Q = 3 : 1

= 3y + y = 180

= 4y = 180

= y = 45

= \(\angle\)Q = \({45}^\circ\)

Average of \(\angle\)R and \(\angle\)Q is

= (\(\angle\)R + \(\angle\)Q)/2

= \(\frac{36\;+\;45}{2}\)

= \(\frac{81}{2}\) = \({40.5}^\circ\)

Therefore, the average of (\(\angle\)R and \(\angle\)Q) is \({40.5}^\circ\).