In a parallelogram ABCDABCD, the bisectors of ∠A∠A and ∠B∠B intersect at SS, ∠B∠B and ∠C∠C at RR, ∠C∠C and ∠D∠D at QQ and ∠D∠D and ∠A∠A at PP. What kind of a quadrilateral is PQRSPQRS?
Answer:
RectangleRectangle
- The situation given in the question is represented by the figure below.
- We are given that ABCDABCD is a parallelogram.
⟹DC∥AB⟹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.