ABCD is a quadrilateral in which bisectors of /_ A and /_ C meet DC produced at Y and BA produced at X respectively. Prove That:
/_X + /_ Y =½( /_ A + /_C).
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Hint-. " /_" symbol means angle
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see diagram.
In the ΔABY: ∠Y + ∠A /2 + ∠B = 180°
In the ΔDCX: ∠D + ∠C /2 + X = 180°
Add them to get: (∠A + ∠C)/2 + ∠B + ∠D + ∠X + ∠Y = 360°
we know ∠A + ∠B + ∠C + ∠D = 360°
From these two equations , we get: ∠X + ∠Y = (∠A+ ∠C) /2
In the ΔABY: ∠Y + ∠A /2 + ∠B = 180°
In the ΔDCX: ∠D + ∠C /2 + X = 180°
Add them to get: (∠A + ∠C)/2 + ∠B + ∠D + ∠X + ∠Y = 360°
we know ∠A + ∠B + ∠C + ∠D = 360°
From these two equations , we get: ∠X + ∠Y = (∠A+ ∠C) /2
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