Question

In below fig, OP, OQ, OR and OS arc four rays. Prove that:
∠POQ + ∠QOR + ∠SOR + ∠POS = 360°

Answer 1

Given that

OP, OQ, OR and OS are four rays

You need to produce any of the ray OP, OQ, OR and OS backwards to a point in the figure. Let us produce ray OQ backwards to a point

T so that TOQ is a line

Ray OP stands on the TOQ

Since `∠`TOP, `∠`POQ is linear pair

`∠`TOP + `∠`POQ = 180°                  .......(1)

Similarly, ray OS stands on the line TOQ

∴`∠`TOS + `∠`SOQ = 180°                      ..........(2)

But `∠`SOQ = `∠`SOR + `∠`QOR

So, (2), becomes

`∠`TOS + `∠`SOR + `∠`OQR = 180°

Now, adding (1) and (3) you get

`∠`TOP + `∠`POQ + `∠`TOS + `∠`SOR + `∠`QOR = 360°

⇒ `∠`TOP + `∠`TOS = `∠`POS

∴ (4) becomes

`∠`POQ + `∠`QOR + `∠`SOR + `∠`POS = 360°