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` .

Explanation

We conclude from the given information that Chords `AP` and `DQ` are joined.

`=>`In chord `AP`,
`=>/_PBA=/_ACP` ..... (Angles on the same segment --- .........`(i)`
Similarly
=> In chord `DQ`,

`=>/_DBQ=/_QCD` ...Angles on the same line segment)....... `(ii)`

`=>ABD` and `PBQ` are line segments intersecting at `B`.

`=>/_PBA=/_DBQ` ...Vertically opposite angles --- `(iii)`
By the equations `(i)`, `(ii)` and `(iii)`,

`/_ACP=/_OCD`
Hence proved.