Question

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

Answer 1

Answer:

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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

Answer 2

Answer:

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Step-by-step explanation:

So ∠ PQB = 90-x = ∠ BQT. In quadrilateral DCRT, ∠DRT = x = ∠DRS. So ∠DSR = 90 -x = ∠DSP. Hence PQRS is a parallelogram.