ABCD is a cyclic quadrilateral.if AP,BP,CR and DR are the angles bisector of angle A,B,C and D respectively.then prove that angle APB + angle CRD = 180 DEGREE.

To prove : ∠APB + ∠ CRD = 180º

In Δs APB and CRD we have

∠ APB + ∠ PAB + ∠ PBA= 180°

and ∠CRD +∠ RCD + ∠RDC=180º

⇒∠ APB +1/2∠ A + 1/2∠ B =180º  [ since AP and BP are bisectors of ∠s A and B ]

and ∠ CRD + 1/2∠ C + 1/2∠ D =180°  [ since DR and CR are bisectors of ∠s D and C ]

∠ APB +1/2∠ A + 1/2∠ B+ ∠ CRD + 1/2∠ C + 1/2∠ D=180º+180°

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

∠ APB+∠ CRD+1/2{180+180}=360°  [ ABCD is cyclic quadrilateral, ∠ A +∠ C=180 and∠ B+∠ D=180]

∠ APB+∠ CRD=360°-180°

∠ APB+∠ CRD=180°