In figure, OP, OQ, OR and OS are four rays. Prove that: + + + =36… - Vidyadip

Question

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

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Answer

Given that $OP, OQ, OR$ and $OS$ are four ray.
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 $\angle\text{TOP},\angle\text{POQ}$ is a linear pair $\angle\text{TOP}+\angle\text{POQ}=180^\circ\dots(1)$ Similarly, Ray $OS$ stands on the line $TOQ$ $\angle\text{TOS}+\angle\text{SOQ}=180^\circ\dots(2)$ But $\angle\text{SOQ}=\angle\text{SOR}+\angle\text{QOR}\dots{(3)}$
So, eq. $(2)$ becomes $\angle\text{TOS}+\angle\text{SOR}+\angle\text{OQR}=180^\circ$
Now, adding $(1)$ and $(3)$ you get: $\angle\text{TOP}+\angle\text{POQ}+\angle\text{TOS}+\angle\text{SOR}+\angle\text{QOR}=360^\circ$
$\angle\text{TOP}+\angle\text{TOS}=\angle\text{POS}$ Equation $(4)$ becomes $\angle\text{POQ}+\angle\text{QOR}+\angle\text{SOR}+\angle\text{POS}=360^\circ$

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