Question

The adjoining figure shows parallelogram ABCD
The bisectors of angles A, B, C and D enclose a
quadrilateral PQRS. Using the exterior angle
theorem prove that each angle of quadrilateral
PQRS is a right angle.
S
R
А​

Answer 1

Answer:

The proof is given below:

∵ ABCD is a parallelogram

∴ AB ║ CD

And, AD ║ BC

We know that if a transverse line cuts two parallel lines, the sum of the interior angles on the same side of the line is 180°

∴ ∠D + ∠A = 180°        (1)

But DQ is bisector of ∠D and AS is angle bisector of ∠A

∴ ∠PDA = ∠D/2

And, ∠PAD = ∠A/2

From (1)

(∠D + ∠A)/2 = 180°/2

or, ∠D/2 + ∠A/2 = 90°

or, ∠PDA + ∠PAD = 90°

In ΔADP

∠SPQ is exterior angle of Δ ADP

∴ ∠SPQ =  ∠PDA + ∠PAD    (∵ exterior angle is equal to the sum of internal angles)

∴ ∠SPQ = 90°

Similarly we can prove that

∠PSR = 90°

∠SRQ = 90°

∠RQP = 90°

Therefore, each angle of quadrilateral PQRS is a right angle.                      (Proved)

thank you