Q. 23.9( 43 Votes )

In figure 3.57, <

Answer :

(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°


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