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. 10.40). Prove that `/_A C P =/_Q C D`.

Answer 1

Given : 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.
To prove: `angle ACP:angle QCD`
Proof: In circle I, `angle ACP=angleABP` (angles in the same segment)....1)
In circle II, ` angle QCD=angle QBD` (angles in the same segment )......(2)
`angle ABP=angle QBD` (vertically opposite angles )
From eqs. (1) and (2), we get `angle ACP=angle QCD`