Question

In the figure, two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R,S and P,Q respectively. Prove that `:` seg SQ `||` seg RP.

Answer 1

Draw seg MN
`square ` MRPN is cyclic and `/_ MNQ ` is its exterior angle.
`/_ MNQ = /_ MRP ` ….(1) …...(Corollary of cyclic quadrilateral theorem )
`square ` MNQS is cyclic
`:. /_ MNQ + /_ MSQ = 180^(@0` …(Theorem of cyclic quadrilateral )
`:. /_ MRP + /_MSQ = 180^(@)` ....[From (1) ]
`:. /_SRP + /_ RSQ = 180^(@)` .....(R-M-S)
`:.` seg `SQ || ` seg `RP ` ....(interior angles test for parallel lines )