We know that l∥m and t is transversal
from the figure we know that ∠APR and ∠PRD are alternate angles
∠APR=∠PRD
We can write it as
1/2∠APR=1/2∠PRD
We know that PS and RQ are the bisectors of ∠APR and
∠PRD
so we get
∠SPR=∠PRQ
hence, PR intersects PS and RQ at points P and R respectively
We get
PS∥RQ
in the same way SR∥PQ
therefore, PQRS is a parallelogram
we know that the interior angles are supplementary
∠BPR+∠PRD=180o
from the figure we know that PQ and RQ are the bisectors of ∠BPR and ∠PRD
We can write it as
2∠QPR+2∠QRP=180o
Dividing the equation by 2
∠QPR+∠QRP=90o.....(1)
Consider △PQR
Using the sum property of triangle
∠PQR+∠QPR+∠QRP=180o
by substituting in equation (1)
∠PQR+90o=180o
∠PQR=90o
We know that PQRS is a parallelogram
it can be written as
∠PQR=∠PSR=90o
We know that the adjacent angles in a parallelogram are supplementary
∠SPQ+90o=180o
∠SPQ=90o
We know that all the interior angles of quadrilateral PQRS are right angles
therefore it is proved that the quadrilateral PQRS are right angles
therefore it is proved that the quadrilateral formed by the bisectors of interior angles is a rectangle.