Given that,

OP, OQ, OR and OS are four rays

You need to produce any of the rays OP, OQ, OR and OS backwards to a point T so that TOQ is a line.

Ray OP stands on line TOQ

∠TOP + ∠POQ = 180^o (Linear pair) (i)

Similarly,

∠TOS + ∠SOQ =180^o(ii)

∠TOS + ∠SOR + ∠OQR = 180^o (iii)

Adding (i) and (iii), we get

∠TOP + ∠POQ + ∠TOS + ∠SOR + ∠QOR = 360^o

∠TOP + ∠TOS = ∠POS

Therefore, ∠POQ + ∠QOR + ∠SOR +∠POS = 360^o.