In figure 3.57, 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

(1) Given PQRS is a cyclic quadrilateral.

∵Opposite angles of a cyclic quadrilateral are supplementary

⇒∠ PSR + ∠ PQR = 180°

⇒∠ PQR = 180° - 110°

⇒∠ PQR = 70°

(2)2 × ∠ PQR = m(arc PR){The measure of an inscribed angle is half the measure of the arc intercepted by it.}

m(arc PR) = 140°

⇒m(arc PQR) = 360° -140° = 220° {Using Measure of a major arc = 360°- measure of its corresponding minor arc}

(3)side PQ ≅ side RQ

∴m(arc PQ) = m(arc RQ){Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent}

⇒m(arc PQR) = m(arc PQ) + m(arc RQ)

⇒m(arc PQR) = 2 × m(arc PQ)

⇒m(arc PQ) = 110°

(4)In ∆ PQR,

∠ PQR + ∠ QRP + ∠ RPQ = 180°{Angle sum property}

⇒∠ PRQ + ∠ RPQ = 180° - ∠ PQR

⇒ 2∠ PRQ = 180° - 70° {∵side PQ ≅ side RQ}

⇒∠ PRQ = 55°