The quadrilateral formed by angle bisectors of a cyclic quadrilateral is :

1. cyclic

2. square

3. rectangle

4. parallelogram

Given,

ABCD is a cyclic quadrilateral in which AP, BP, CR and DR are the angle bisectors of ∠A, ∠B, ∠C and ∠D respectively, forming quadrilateral PQRS.

In ΔPAB,

∠APB + ∠PAB + ∠PBA = 180° [Sum of the angles of a triangle is 180°]

But,

∠PAB = $\dfrac{1}{2}$∠A and ∠PBA = $\dfrac{1}{2}$∠B [AP and BP are angle bisectors]

So,

∠APB + $\dfrac{1}{2}$∠A + $\dfrac{1}{2}$∠B = 180° …(i)

Similarly, in ΔRCD,

∠CRD + ∠RCD + ∠RDC = 180° [Sum of the angles of a triangle is 180°]

But,

∠RCD = $\dfrac{1}{2}$∠C and ∠RDC = $\dfrac{1}{2}$∠D [CR and DR are angle bisectors]

So,

∠CRD + $\dfrac{1}{2}$∠C + $\dfrac{1}{2}$∠D = 180° …(ii)

Adding (i) and (ii),

∠APB + ∠CRD + $\dfrac{1}{2}$ (∠A + ∠B + ∠C + ∠D) = 360°

But,

∠A + ∠B + ∠C + ∠D = 360° [Sum of angles of a quadrilateral]

So,

∠APB + ∠CRD + $\dfrac{1}{2}$ × 360° = 360°

∠APB + ∠CRD + 180° = 360°

∠APB + ∠CRD = 180°

Since a pair of opposite angles of quadrilateral PQRS is supplementary,

PQRS is a cyclic quadrilateral.

Hence, option 1 is the correct option.