Question

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. 9.27). Prove that ∠ACP = ∠ QCD

Answer 1

Join chords∠ AP and ∠DQ.
For chord AP,
∠PBA=∠ACP (Angles in the same segment) .... (1)
For chord DQ,
∠DBQ=∠QCD (Angles in the same segment) …. (2)
ABD and PBQ are line segments intersecting at B.

∠PBA=∠DBQ (Vertically opposite angles) .... (3)
From equations (1), (2), and (3), we obtain

∠ACP=∠QCD