Q23 of 59 Page 83

In figure 3.101, 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.




We join MN.


As PRMN is a cyclic quadrilateral,


∠R + ∠PNM = 180° …………………..(1) (opposite angles of a cyclic quadrilateral)


Also, QSMN is a cyclic quadrilateral,


∠S + ∠ QNM = 180° ……………………(2) (opposite angles of a cyclic quadrilateral)


Adding (1) and (2)


∠ R + ∠S + ∠ PNM + ∠QNM = 360°


⇒ ∠ R + ∠S + 180 = 360 (PQ is a straight line)


⇒ ∠ R + ∠S = 180°


Similarly we have,


As PRMN is a cyclic quadrilateral,


∠P + ∠RMN = 180° …………………..(3) (opposite angles of a cyclic quadrilateral)


Also, QSMN is a cyclic quadrilateral,


∠Q + ∠ SMN = 180° ……………………(4) (opposite angles of a cyclic quadrilateral)


Adding (3) and (4)


∠ P + ∠Q + ∠ RMN + ∠SMN = 360°


⇒ ∠ P + ∠Q + 180 = 360 (RS is a straight line)


⇒ ∠ P + ∠Q = 180°


Therefore, PR ∥ SQ.


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