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 Fig. 10.40). Prove that∠ ACP = ∠ QCD

Join chords AP and DQ

For chord AP,

∠PBA = ∠ACP (Angles in the same segment) (i)

For chord DQ,

∠DBQ = ∠QCD (Angles in the same segment) (ii)

ABD and PBQ are line segments intersecting at B

∠PBA = ∠DBQ (Vertically opposite angles) (iii)

From (i), (ii), and (iii), we get

∠ACP = ∠QCD