Question

pqrs is a parallelogram and A is the midpoint of PS .QA and RS are produced to meet at B. prove that BR is equal to 2 PQ​

Answer 1

Answer:We have a parallelogram PQRS , And PQ = RS , PQ | | RS and QR = SP , QR | | SP ,

And PQ = 2 PS , and M is mid pint of SR , So , we can say

QR = SP = RM = MS ,

So from base angle theorem In ∆PMS we can say that

∠ PMS = ∠ MPS = ∠ x And

In ∆QMR we can say that

∠ QMR = ∠ MQR = ∠ y ,

We know adjacent angles are supplementary in parallelogram , So

∠ R + ∠ S = 180° ----------- ( 1 )

And from angle sum property in ∆ PMS , we know

∠ S + ∠ x + ∠ x = 180°

∠ S + 2∠ x = 180° ----------- ( 2 )

And in ∆ QMR , we know

∠ R + ∠ y + ∠ y = 180°

∠ R + 2∠ y = 180° ----------- ( 3 )

Add equation 2 and 3 we get

∠ S + ∠ R + 2 ∠ x + 2 ∠ y = 360° , Substitute value from equation 1 , we get

180°+ 2 ∠ x + 2 ∠ y = 360°

2 ∠ x + 2 ∠ y = 180°

∠ x + ∠ y = 90° ----------- ( 4 )

And we know

∠ PMQ + ∠ x + ∠ y = 180° ( As these are the linear angles form on a straight line RS )

Substitute value from equation 4 , we get

∠ PMQ + 90° = 180°

∠ PMQ = 90° ( Hence proved )

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