Question

2 circles cut the point A and B. QS is the line drawn through B prove that angle APQ and angle ARS = 180°​

Answer 1

Join AB and let XY be the tangent at P. Then by alternate segment theorem,

∠APX=∠ABP ……………(i)

Next, ABCD is a cyclic quadrilateral, therefore, by the theorem sum of the opposite angles of a quadrilateral is 180^{\circ}

∠ABD+∠ACD=180  

Also,  ∠ABD=∠ABP=180  

 (Linear Pair)

∴∠ACD=∠ABP ...........(ii)

From (i) and (ii),

∠ACD=∠APX

∴XY∥CD    (Since alternate angles are equal).