Math, asked by Ashutosh19821982, 1 month ago

In traingle ABC, angle C = 40°. BQ and AR are the angle bisectors of angle B and angle A, respectively. Find traingle APQ.​

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Answered by shkmaria2006
0

Answer:

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Given△ABCisinscribedinC(0,r).

Thebisectorsof∠BAC,∠ABCand∠ACBmeetsthecircumcircle

of△ABC,inP,Q,Rrespectively.

InthefigureJoinRQ,

∠ABQ=∠APQ−(i)

{Anglesinthesamesegmentofacircleareequal}.

∠ABQ=∠QBC{BQisthebisectorof∠ABC}.

∴∠QBC=∠APQ−(ii)

Adding(i)&(ii)

∠ABQ+∠QBC=∠APQ+∠APQ

∴∠ABC=2∠APQ−(iii)

Similarily,∠ACB=2∠APR−(iv)

Adding(iii)&(iv)

∠ABC+∠ACB=2(∠APQ+∠APR)

∴∠ABC+∠ACB=2∠QPR−(v)

In△ABC,

∠ABC+∠BAC+∠ACB=180

{Anglesumproperty}

∠ABC+∠ACB=180

−∠BAC−(vi)

From(v)&(vi),weget

180

−∠BAC=2∠QPR

∴∠QPR=

2

1

(180−∠BAC)

∠QPR=

2

1

×180

2

1

∠BAC

⟹∠QPR=90−

2

1

∠BAC

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