Question

in given figure angle poq =100 and angle pqr=30,then find angle rpo

pls answer

Answer 1

∠RPO =    60° if in Fig ∠poq =100 and ∠pqr=30

Step-by-step explanation:

∠PRQ  = (1/2) POQ   ( Angle subtended by same chord)

=> ∠PRQ = (1/2) 100°

=> ∠PRQ = 50°

in Δ PQR

∠PRQ  + ∠PQR  + ∠RPQ = 180°

=> 50° + 30° + ∠RPQ = 180°

=> ∠RPQ = 100°

in Δ PQQ

∠OPQ = ∠OQP   as OP = OQ  ( Radius)

∠OPQ + ∠OQP + ∠POQ = 180°

=> 2∠OPQ + 100° = 180°

=> ∠OPQ = 40°

∠RPO =    ∠RPQ - ∠OPQ

=> ∠RPO =    100° -  40°

=> ∠RPO =    60°

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Answer 2

∠RPO = 60°

Step-by-step explanation:

In the given figure,

∠POQ = 100° and ∠PQR = 30°

$\angle PRQ = \dfrac{1}{2}\angle POQ$

∵ If angle subtended on same arc then central angle is double angle at circle.

$\angle PRQ = \dfrac{1}{2}\times 100^\circ$

$\angle PRQ = 50^\circ$

In ΔPOQ, PO = QO (radius of same circle)

Therefore, POQ is isosceles triangle.

$\angle OPQ = \angle OQP$

In ΔPOQ

$\angle OPQ+ \angle OQP+\angle POQ=180^\circ$  (angle sum property of triangle)

$2\angle OPQ+ 100^\circ=180^\circ$

$\angle OPQ=40^\circ$

In ΔPQR

$\angle PQR + \angle PRQ+\angle QPR=180^\circ$

$30^\circ+50^\circ+(\angle RPO+40^\circ)=180^\circ$

$\angle RPO = 180^\circ-30^\circ-50^\circ-40^\circ$

$\angle RPO = 60^\circ$

Hence, the measure of angle RPO is 60°

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