# If bisectors of < Sum of all angles of a quadrilateral is 360°

A + B + C + D = 360°

On dividing both sides by 2,

1/2(A + B + C + D) = 1/2 × 360° = 180°

AP, PB, RC and RD are bisectors of A, B, C and D

PAB + ABB + RCD + RDC = 180° …(1)

Sum of all angles of a triangle is 180°

∴ ∠PAB + APB + ABP = 180°

PAB + ABP = 180° – APB …(2)

Similarly,

∴ ∠RDC + RCD + CRD = 180°

RDC + RCD = 180° – CRD …(3)

Putting (2) and (3) in (1),

180° – APB + 180° – CRD = 180°

360° – APB CRD = 180°

APB + CRD = 360° – 180°

APB + CRD = 180° …(4)

Now,

SPQ = APB [vertically opposite angles]

SRQ = DRC [vertically opposite angles]

Putting in (4),

⇒ ∠SPQ + SRQ = 180°

Hence, PQRS is a quadrilateral whose opposite angles are supplementary.

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