Question

In fig $OP,OQ,ON$ and $OS$ are four rays.
Prove that:
$\angle POQ+\angle QOR+\angle ROS+\angle SOP={360}^o$

Answer 1

Lets extend ray PQ backwards till T

Now,

$\angle TOP+\angle POQ={180}^o$ (Linear pair) ………$\left(1\right)$

Also,

$\angle TOS+\angle SOR+\angle QOR={180}^o$ (Linear pair) ……..$\left(2\right)$

Adding $\left(1\right)$ & $\left(2\right)$

$\angle TOP+\angle POQ+\angle TOS+\angle SOR+\angle QOR={360}^o$

But $\angle TOP+\angle TOS=\angle POS$

$\angle POQ+\angle QOR+\angle SOR+\angle POS={360}^o$.