Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.
Solution.
Given: Let ABCD be a parallelogram and AP, BR, CQ, DS are the bisectors of
and respectively.
To Prove: Quadrilateral PQRS is a rectangle.
Proof: Since ABCD is a parallelogram
Then and DA is transversal.
{sum of co-interior angles of a parallelogram}
{Dividing both sides by 2}
{ sum of all the angles of a triangle is 1800}
Similarly,
Similarly,
{ sum of all the angles of a triangle is 1800}
{ vertically opposite angles}
Similarly,|
{ sum of all the angles of a triangle is 1800}
{ vertically opposite angles}
Thus PQRS is a quadrilateral whose all angles are
Hence PQRS is a rectangle.
Hence proved