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प्रश्न
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.
उत्तर
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.
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संबंधित प्रश्न
Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”.
Prove that “The opposite angles of a cyclic quadrilateral are supplementary”.
In the given figure, ▢PQRS is cyclic. side PQ ≅ side RQ. ∠PSR = 110°, Find -
(1) measure of ∠PQR
(2) m(arc PQR)
(3) m(arc QR)
(4) measure of ∠PRQ
`square`MRPN is cyclic, ∠R = (5x – 13)°, ∠N = (4x + 4)°. Find measures of ∠R and ∠N.
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(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.
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MRPN is cyclic, ∠R = (5x – 13)°, ∠N = (4x + 4)°. Find measures of ∠R and ∠N, by completing the following activity.
Solution:
MRPN is cyclic
The opposite angles of a cyclic square are `square`
∠R + ∠N = `square`
∴ (5x – 13)° + (4x + 4)° = `square`
∴ 9x = 189°
∴ x = `square`
∴ ∠R = (5x – 13)° = `square`
∴ ∠N = (4x + 4)° = `square`
Prove the following theorems:
Opposite angles of a cyclic quadrilateral are supplementary.
In the figure, PQRS is cyclic, side PQ ≅ side RQ, ∠PSR = 110°. Find
(i) measure of ∠PQR
(ii) m(arc PQR)
(iii) m(arc QR)
