Question

in the figure PQRS is a cyclic side PQ is congruent to RQ , angle PSR = 110 find measure of angle PQR , m ( PQR ) , m ( arc QR ) , m angle PRQ​

Answer 1

Answer:

Step-by-step explanation:

(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