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