In a parallelogram ABCD, the bisectors of ∠A and ∠B intersect at S, ∠B and ∠C at R, ∠C and ∠D at Q and ∠D and ∠A at P. What kind of a quadrilateral is PQRS?
Answer:
Rectangle
- The situation given in the question is represented by the figure below.
- We are given that ABCD is a parallelogram.
⟹DC∥AB
Also, as the adjacent angles of a parallelogram are supplementary, we have
∠A+∠D=180∘⟹12∠A+12∠D=90∘⟹∠PAD+∠ADP=90∘⟹∠APD=90∘[Sum of angles of a triangle is 180∘.]⟹∠SPQ=90∘[Vertically opposite angles.]
Similarly, ∠PQR=90∘,∠QRS=90∘, and ∠PSR=90∘.
Thus, PQRS is a quadrilateral each of whose angles is 90∘.
Hence, PQRS is a rectangle.