Abcd Quadrilateral In Which Bisectors And Meet Produced And Ba Produced At Respectively

ABCD is a quadrilateral in which the bisectors of ∠A and ∠C meet DC produced at Y and BA produced at X respectively. Prove that .

Answers:

∠X + ∠Y = 1/2 (∠A + ∠C)

Here, ∠1 = ∠2 and ∠3 and ∠4 

In △XBC , we have 

∠X + ∠B + ∠4  = 180°  

∠X + ∠B + 1/2 ∠C = 180°  

In △ADY , we have 

∠2 + ∠D +∠Y = 180°   

1 / 2 + ∠A + ∠D + ∠Y = 180°  

Adding (i) and (ii) , we have    

 ∠X + ∠Y + ∠B + ∠D + 1 / 2 ∠C + 1 / 2 ∠A = 360°  

Also, in quadrilateral ABCD,

 ∠A + ∠B + ∠C + ∠D = 360° 

∴ ∠X + ∠Y + ∠B + ∠D + 1 / 2 ∠C  + 1 / 2 ∠A  =  ∠A + ∠B + ∠C + ∠D  

∠X + ∠Y = ∠A – 1 / 2 ∠A + ∠C – 1 / 2 ∠C 

∠X + ∠Y = 1 / 2 ( ∠A + ∠C)