Question

In Fig. 6.11, OP, OQ, OR and OS are four rays. Prove that Z POQ + Z QOR + Z SOR + Z POS = 360° ??



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Answer 1

Question :-

In Fig. 6.11, OP, OQ, OR and OS are four rays. Prove that Z POQ + Z QOR + Z SOR + Z POS = 360° ?

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Given :-

• op,oq,or,os are four rays

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To prove :-

• Prove that Z POQ + Z QOR + Z SOR + Z POS = 360°

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explanation :-

in fig 6.11 you need to produce any of the rays op,oq,or,or os backwards to point.

let us produce ray oq backwards to a point t so that toq is a line u can check in figure

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Solution:-

now Ray op stands on line toq therefore,

$\rm :\longmapsto\: \angle \: top + \angle \: poq = 180 \degree - - - - (1)$

similarly,ray os stands on line toq therefore,

$\rm :\longmapsto\: \angle \: tos + \angle \: soq = 180 \degree - - - - (2)$

but,

$\rm :\longmapsto\: \angle \: soq = \angle \: sor + \angle \: qor$

so (2) becomes,

$\rm :\longmapsto\: \angle \: tos + \angle \: sor + \angle \: qor = 180 \degree$

now by adding (1) and (3) we get,

$\rm :\longmapsto\: \angle \: tos + \angle \: poq + \angle \: tos + \angle \: sor + \angle \: qor \: = 360 \degree$

but,

$\rm :\longmapsto\: \angle \: top + \angle \: tos = \angle \: pos$

therefore (4) becomes

$\rm :\longmapsto\: \angle \: poq + \angle \: qor + \angle sor + \angle \: pos = 360 \degree$

hence proved

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