Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that ∠ACP = ∠QCD.

 proof:-  segments AD&PQ intersect at B

so, angle ABP= angleQBD ( verticaly opposite angles).  -(i)

in major circle, angle subtended by minor arc AP = angle ACP= angle ABP      -(ii)

in minor circle, angle subtended by minor arc PQ= angleQCD= angleQBD.        -(iii)

from (i) ,(ii) and (iii) 

angle ACP= angle QCD