Question

OP,OQ,OR and OS are four rays Prove that Angle POQ + Angle QOR + Angle SOR + Angle POS =360°​

Answer 1

Step-by-step explanation:

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Answer 2

Solution:-

you need to produce any of the rays OP,OQ,OR or OS. backwards to a point. let's produce a ray OQ backwards to a point T so that TOQ is a line .

Now ray OP stands on the line TOQ .

⠀∴ ⠀⠀⠀⠀⠀∠TOP + ∠SOQ = 180 (1)

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀(linear pair )

Similarly ray OS stands on line TOQ.

∴⠀⠀⠀⠀⠀⠀⠀ ∠TOS + ∠ SOQ = 180° (2)

but ⠀⠀⠀⠀⠀ ∠SOQ = ∠SOR + ∠QOR

So (2) becomes :-

⠀⠀⠀⠀⠀⠀∠TOS + ∠SOR + ∠QOR =180°

Now adding (1) and (2) we get :-

∠TOP + ∠POQ + ∠TOS + ∠SOR + ∠QOR ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀=360°

But ⠀⠀⠀⠀⠀⠀ ∠TOP + ∠TOS = ∠POS

Therefore it becomes :-

∠POQ + ∠QOR+ ∠SOR + ∠POS = 360°

Hence proved √