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प्रश्न
If two consecutive angles of cyclic quadrilateral are congruent, then prove that one pair of opposite sides is congruent and other is parallel.
उत्तर
Given: ABCD is a cyclic quadrilateral and ∠ABC ≅ ∠BCD.
To prove: Side DC ≅ Side AB, AD || BC
Construction: Draw seg AM and seg DN both perpendicular to side BC.
Proof: ∠ABC ≅ ∠BCD ......(i) [Given]
∠ABC + ∠ADC = 180° ......(ii) [Opposite angles of a cyclic quadrilateral are supplementary]
From equations (i) and (ii),
∠BCD + ∠ADC = 180°
∴ Side AD || Side BC .....[Interior angles test]
In ΔDNC and ΔAMB,
seg DN ≅ seg AM .......[Perpendicular distance between two parallel lines]
∠DNC ≅ ∠AMB ......[Each is 90°]
∠DCN ≅ ∠ABM ......[Given]
As a result, the SAA test of congruence
ΔDNC ≅ ΔAMB
∴ Side DC ≅ Side AB ......[C.S.C.T.]
Hence, side AD || side BC and side DC ≅ side AB.
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.
In the given figure, line PR touches the circle at point Q. Answer the following questions with the help of the figure.
(1) What is the sum of ∠ TAQ and ∠ TSQ ?
(2) Find the angles which are congruent to ∠ AQP.
(3) Which angles are congruent to ∠ QTS ?
(4) ∠TAS = 65°, find the measure of ∠TQS and arc TS.
(5) If ∠AQP = 42°and ∠SQR = 58° find measure of ∠ATS.
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.
In the given figure, two circles intersect each other at points A and E. Their common secant through E intersects the circles at points B and D. The tangents of the circles at points B and D intersect each other at point C. Prove that ▢ABCD is cyclic.
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Find the value of x in the given figure.
A school wants to conduct tree plantation programme. For this a teacher allotted a circle of radius 6 m ground to nineth standard students for planting sapplings. Four students plant trees at the points A, B, C and D as shown in figure. Here AB = 8 m, CD = 10 m and AB ⊥ CD. If another student places a flower pot at the point P, the intersection of AB and CD, then find the distance from the centre to P.
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)
