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Question

If the angle bisectors of a parallelogram form another quadrilateral, then this quadrilateral will be a 


  1. Square

  2. Rhombus

  3. Parallelogram

  4. rectangle


Solution

The correct option is D

rectangle


If the angle bisector of a parallelogram form another quadrilateral, then this quadrilateral will be a rectangle
Given:- ABCD is a parellelogram, AP, BR, CR and DP are angle bisectors.
To prove:- PQRS is a rectangle.
Proof:- In parellelogram ABCD, ADCB
Therefore A + B=180. [ Sum of Angles on the  same side of transversal is 180]
  12( A + B)= (180)2
  12( A + B)=90 .....(1)
By Angle Sum Property in ASB,
ABS + BSA + SAB= 180
  12 B + BSA + 12  A= 180( given that AP and BR are angle bisectors)
90+ BSA =180 (From (1))
BSA = 90
Similarly , BRC= CQD= APD =90
A quadrilateral PQRS in which all angles are right angles is a rectangle. 

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