Question

What is the quadrilateral formed by the angle bisectors of a parallelogram?

(a) Square

(b) Rectangle

(c) Circle

(d) Rhombus

I got this question in semester exam.

This intriguing question comes from Properties of a Parallelogram in section Quadrilaterals of Mathematics – Class 9

Answer 1

Correct choice is (b) Rectangle

For explanation I would say: PQ || RS ⇒ ∠P + ∠S = 180°  (Interior angles on the same side of transversal)

⇒ ½ ∠P + ½ ∠S = ½ x 180°

⇒ ∠APS + ∠PSA = 90°  ——————-(i)

In ΔASP, ∠APS + ∠PSA + ∠SAP = 180°  (Angle sum property of triangle)

⇒ 90° + ∠SAP = 180°  (from equation i)

⇒ ∠SAP = 90°

⇒ ∠DAB = 90° (Vertically opposite angle)

Similarly, ∠ADB = 90°, ∠ABC = 90°, ∠DCB = 90°.

Hence, ABCD is a rectangle.