Question

Question 9
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 $\angle ACP=\angle QCD$

Answer 1

Construction:
Join chords AP and DQ.
For chord AP,
$\angle PBA=\angle ACP$ (Angles in the same segment) ... (1)
For chord DQ,
$\angle DBQ=\angle QCD$ (Angles in the same segment) ... (2)
ABD and PBQ are line segments intersecting at B.
$\therefore \angle PBA=\angle DBQ$ (Vertically opposite angles) ... (3)
From equations (1), (2), and (3), we obtain
$\angle ACP=\angle QCD$