Question

.In the figure quadrilateral PQRS is cyclic./PSR=120⁰.The measure of /PQR and arc PQR respectively are ____. *

Answer 1

(1) Given PQRS is a cyclic quadrilateral. ∴ Opposite angles of a cyclic quadrilateral are supplementary

⇒∠PSR+∠PQR=180

o

⇒∠PQR=180

o

−110

o

⇒∠PQR=70

o

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

m(arcPR)=140

o

⇒m(arcPQR)=360

o

−140

o

=220

o

{Using Measure of a major arc = 360

o

- measure of its corresponding minor arc}

(3)side PQ≅ side RQ

∴m(arcPQ)=m(arcRQ) {Corresponding arcs of congruent chords of a circle (or congruent circles) are congruent} ⇒m(arcPQR)=m(arcPQ)+m(arcRQ)

⇒m(arcPQR)=2×m(arcPQ)

⇒m(arcPQ)=110

o

(4)In △PQR,

∠PQR+∠QRP+∠RPQ=180

o

{Angle sum property}

⇒∠PRQ+∠RPQ=180

o

−PQR

⇒2∠PRQ=180

o

−70

o

{∴sidePQ≅sideRQ}

⇒∠PRQ=55

o

i hope it's helpful to u. ❤