ABCD is a quadrilateral in which the bisectors of angleA angleC meet DC produced at Y and BA produced at Y respectively. Prove that AngleX + AngleY = 1/2 (angleA+AngleC) Please answer this question urgently. My exam is on 20th Thursday. Thanks Share with your friends Share 2 Utsav answered this Given ABCD is a quadrilateral in which the AX bisects ∠DAB and CY bisects ∠BCD . To prove ∠X+∠Y =12(∠A+∠C) Now as we know that sum of interior angles of a quadrilateral is 360°Hence ∠DAB+∠ABC+∠BCD+∠ADC = 360°⇒∠A+∠B+∠C+∠D = 360°...............(1)Now as we know that sum of interior angles of a triangle is 180°Hence in ∆ADX , we have ∠ADC+∠DXA+∠XAD = 180°⇒∠D+∠X+∠A2=180°............(2)Similarly in ∆CYB,we have ∠CYB+∠YCB+∠CBY = 180°⇒∠Y+∠C2+∠B=180°...............(3)Adding (2) and (3), we get ∠D+∠X+∠A2+∠Y+∠C2+∠B=360° ..........(4)Equating LHS of (1) and (4) as RHS of (1) and (4) are same ∠D+∠X+∠A2+∠Y+∠C2+∠B=∠A+∠B+∠C+∠D ⇒∠X+∠Y =∠A-∠A2+∠C-∠C2⇒∠X+∠Y =∠A2+∠C2Hence ∠X+∠Y =12(∠A+∠C) proved 2 View Full Answer
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