Question

the bisectors of angles of a quadrilateral enclose a rectangle, then show that it is a parallelogram.​

Answer 1

Answer:Quadrilateral PQRS has angle bisectors PT,QA,RA,SC.

ΔPQB,ΔQBT,ΔSDC are right angled triangle.

Let angle P=2x

so, ∠PQB=90−x=∠BQT

∴∠QTB=(90−(90−x))=x

∠CTR=180−x

In triangle SDR,

∠RDS=90  

∘

, in parallelogram DCTR

∠DCT & ∠CDR=90  

∘

 

∴∠DRT=x & ∠DRS=x

∴∠DSR=90−x

sum of adjacent angles, ∠P+∠Q=180  

∘

 

Opposite angles ∠P=∠R,∠Q=∠C

∴ PQRS is parallelogram