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प्रश्न
Prove that the “the opposite angles of the cyclic quadrilateral are supplementary”.
उत्तर
arc ABC is intercepted by the inscribed angle ∠ADC.
∴ ∠ADC = `1/2` m(arc ABC) ......(i) [Inscribed angle theorem]
Similarly, ∠ABC is an inscribed angle.
It intercepts arc ADC.
∴ ∠ABC =`1/2` m(arc ADC) ......(ii) [Inscribed angle theorem]
∴ ∠ADC + ∠ABC = `1/2` m(arc ABC) + `1/2` m(arc ADC) ....[Adding (i) and (ii)]
∴ ∠D + ∠B = `1/2` m (arc ABC) + m(arc ADC)]
∴ ∠B + ∠D = `1/2 xx 360^circ` ......[arc ABC and arc ADC constitute a complete circle]
= 180°
∴ ∠B + ∠D = 180°
Similarly we can prove,
∠A + ∠C = 180°
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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)
