Question

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

Answer 1

In the figure, OP, OQ, OR and OS are the rays from O
To prove: $\angle POQ+\angle QOR+\angle SOR+\angle POS={360}^{\circ }$
Construction: Produce PO to E

Proof: $\angle POQ+\angle QOE={180}^{\circ }$
(Linear pair)
Similarly, $\angle EOS+\angle POS={180}^{\circ }$
Adding we get,
$\angle POQ+\angle QOR+\angle ROE+\angle EOS+\angle POS={180}^{\circ }+{180}^{\circ }\Rightarrow \angle POQ+\angle QOR+\angle ROS+\angle POS={360}^{\circ }$
Hence $\angle POQ+\angle QOR+\angle SOR+\angle POS={360}^{\circ }$