Question
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.
Prove that :seg SQ || seg RP.
✓
Answer
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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