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If bisectors of angle A and angle B of a...

If bisectors of `angle A and angle B` of a quadrilateral ABCD intersect each other at P, of `angle B and angle C " at " Q, " of " angle C and angle D " at R and of " angle D and angle A ` at S then PQRS is a

A

rectangle

B

parallelogram

C

rhombus

D

quardilateral whose opposite angles are supplementary

Text Solution

Verified by Experts

The correct Answer is:
D
`angle APB =180^(@)-((1)/(2)angle A + (1)/(2) angle B).`
`angle CRD =180^(@) -((1)/(2)angle C +(1)/(2) angle D).`
`therefore angle SPQ + angle SRQ =angle APB + angle CRD`
`=360^(@)-(1)/(2) (angle A +angle B+angle C+angle D)`
`=360^(@) -180^(@) =180^(@).`
Now, `angle PSR + angle PQR =360^(@)-(angle SPQ+angle SRQ)`
`=360^(@)-180^(@)=180^(@).`
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Knowledge Check

  • If bisectors of angleA and angleB of a quadrilateral ABCD intersect each other at P, of angleB and angleC at Q, of angleC and angleD at R and of angleD and angleA at S, then PQRS is a

    A
    rectangle
    B
    rhombus
    C
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  • If the bisector of the angles A and B of a quadrilateral ABCD meet at O , then angle AOB is equal to:

    A
    `angle C + angle D`
    B
    `(1)/(2) ( angle C + angle D)`
    C
    `(1)/(2) angle C + (1)/(3) angle D`
    D
    `(1)/(3) angle C + (1)/(2) angle D`

  • Diagonals of a quadrilateral bisect each other. If angle A = 45^(@) , then angle B equals

    A
    `45^(@)`
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    `55^(@)`
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    `135^(@)`
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