Question

In the given 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

It is given that two circles intersect at points M and N. Secants drawn through M and N intersect the circles at points R, S and P, Q.
Join MN. 

Quadrilateral PRMN is a cyclic quadrilateral.
∴ ∠PRM = ∠MNQ       .....(1)           (Exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle)
Quadrilateral QSMN is a cyclic quadrilateral.
∴ ∠QSM = ∠MNP       .....(2)           (Exterior angle of a cyclic quadrilateral is congruent to the angle opposite to its adjacent interior angle)
Adding (1) and (2), we get
∠PRM + ∠QSM = ∠MNQ + ∠MNP              .....(3)
Now, 
∠MNQ + ∠MNP = 180º      .....(4)        (Angles in linear pair)   
From (3) and (4), we get
∠PRM + ∠QSM = 180º      
Now, line RS is transversal to the lines PR and QS such that 
∠PRS + ∠QSR = 180º
∴ seg SQ || seg RP      (If the interior angles formed by a transversal of two distinct lines are supplementary, then the two lines are parallel)

Hence proved.