Question

In figure $OP,OQ,OR$ and $OS$ are four rays. Prove that $\angle POQ+\angle QOR+\angle SOR+\angle POS={360}^o$

Answer 1

Let extend ray $OQ$ backwards till $T$

$\Rightarrow$ Now, $\angle TOP+\angle POQ={180}^o$ [ Linear pair ] --- ( 1 )

$\Rightarrow$ $\angle TOS+\angle SOR+\angle QOR={180}^o$ [ Linear pair ] ----- ( 2 )

Adding equation ( 1 ) and ( 2 ),

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

Since, $\angle TOP+\angle TOS=\angle POS$

$\therefore$ $\angle POQ+\angle OR+\angle SOR+\angle POS={360}^o$ --- Hence proved