Question

pqrs is such quadrilateral that p is the centre of circle passing through q p s prove that rqs + rsq= 1/2 qps

Answer 1

Here is ur ans:-
Here we know Arc QR subtends ∠ QPR at the center and ∠ RSQ at a point " S " in the remaining part of circle . So,

∠ QPR = 2 ∠ RSQ ------------- ( 1 )

And

we know Arc RS subtends ∠ SPR at the center and ∠ RQS at a point " Q " in the remaining part of circle . So,

∠ SPR = 2 ∠ RQS ------------- ( 2 )

Now we add equation 1 and 2 , we get

∠ QPR + ∠ SPR = 2 ∠ RSQ + 2 ∠ RQS

2 ( ∠ RQS + ∠ RSQ ) = ∠ QPS ( As we know ∠ QPR + ∠ SPR = ∠ QPS )

So,

∠ RQS + ∠ RSQ = ∠ QPS2 ( Hence proved )
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Answer 2

hii friend!

arc QR subtends angle QPR at the centre
∠ RSQ at a point S  in the remaining part of the circle .

∠ QPR = 2 ∠ RSQ -------- ( marked as 1 )


 Arc RS subtends ∠ SPR at the centre 
 ∠ RQS at a point  Q  in the remaining part of the circle . 


∠ SPR = 2 ∠ RQS   --------- ( marked as 2 )

by adding equation 1 and 2, 

∠ QPR + ∠ SPR
= 2 ∠ RSQ + 2 ∠ RQS
= 2 ( ∠ RQS + ∠ RSQ )
= ∠ QPS (∵∠ QPR + ∠ SPR = ∠ QPS )

∠ RQS + ∠ RSQ = ∠ QPS2 (  proved )

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