Question

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

Answer 1

(1)
∠PSR + ∠PQR = 180º (Opposite angles of a cyclic quadrilateral are supplementary)

⇒ 110º + ∠PQR = 180º

⇒ ∠PQR = 180º − 110º = 70º

Thus, the measure of ∠PQR is 70º.

(2)
∠PSR = $\frac{1}{2}$ m(arc PQR) (Measure of an inscribed angle is half of the measure of the arc intercepted by it)

⇒ m(arc PQR) = 2 × ∠PSR = 2 × 110º = 220º

(3)
side PQ ≅ side RQ

⇒ arc PQ ≅ arc RQ

⇒ m(arc PQ) = m(arc RQ) .....(1)

Now,

m(arc PQR) = m(arc PQ) + m(arc QR)

⇒ m(arc PQR) = m(arc QR) + m(arc QR) [From (1)]

⇒ m(arc PQR) = 2 × m(arc QR)

⇒ m(arc QR) = $\frac{220°}{2}$ = 110º [m(arc PQR) = 220º]

(4)
​∠PRQ = $\frac{1}{2}$m(arc QR) (Measure of an inscribed angle is half of the measure of the arc intercepted by it)

⇒ ∠PRQ = $\frac{1}{2}\times 110°$ = 55º [m(arc QR) = 110º]