Question

Show that the bisectors of angles of a parallelogram form a rectangle

Answer 1

Given: A parallelogram in which bisector of angle A, B, C, D intersect at P, Q, R, S to form a quadrilateral PQRS.

To prove: Quadrilateral PQRS is a rectangle.

Proof: Since ABCD is a parallelogram.

Therefore, AB || DC.

Now, AB || DC, and transversal AD cuts them, so we have

∠A + ∠D = 180°

`1/2 ∠"A" + 1/2 ∠ "D" = (180^circ)/2`

∠DAS + ∠ADS = 90°

But in ΔASD, we have

∠ADS + ∠DAS + ∠ASD = 180°

90° + ∠ASD = 180°

∠ASD = 90°

∠RSP = ∠ASD ...(vertically opposite angle)

∠RSP = 90°

Similarly, we can prove that

∠SRQ = 90°, ∠RQP = 90° and ∠QPS = 90°

Thus, PQRS is a quadrilateral each of whose angle is 90°.

Hence, PQRS is a rectangle.