Question

In figure 3.101. two circles intersect
at points M and N. Secant
drawn through M and N intersect
the circles at points R, S and
P, Q respectively.
Prove that : seg SQ Il seg RP.​

Answer 1

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.

Hope it helps you ❣️☑️☑️☑️

Answer 2

Hope this will help you