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Question 13
Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

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Solution

Given Let ABCD be a parallelogram and AP, BR, CR, be are the bisectors of A,B,C and D, respectively.

To prove Quadrlateral PQRS is a rectangle.

Proof Since, ABCD is a parallelogram, then DC||AB and DA is a transversal.
We have, A+D=180
[sum of cointerior angles of a parallelogram is 180]
12A+12D=90 [dividing both sides by 2]
PAD+PDA=90
APD=90 [since, sum of all angles of a triangle is 180]
SPQ=90 [vertically opposite angles]
Similarly, PQR=90
QRS=90
PSR=90

Thus, PQRS is a quadrilateral whose each angle is 90.

Hence, PQRS is a rectangle.


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