Question

In the adjoining 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

Given: Two circles intersect each other at points M and N. 

To prove: seg SQ || seg RP 

Construction: Join seg MN. 

Proof: 

□RMNP is a cyclic quadrilateral. 

∴ ∠MRP = ∠MNQ …………. (i) [Corollary of cyclic quadrilateral theorem] 

Also, □MNQS is a cyclic quadrilateral.

∴ ∠MNQ+ ∠MSQ = 180° [Theorem of cyclic quadrilateral] 

∴ ∠MRP + ∠MSQ = 180° [From (i)]

But, they are a pair of interior angles on the sarpe side of transversal RS on lines SQ and RP. 

∴ seg SQ || seg RP [Interior angles test]