Question

in fig the side QR of triangle PQR is produced to a point S .if the bisectors of angle PQR and angle PRS meet at point T then prove that angle QTR =1/2angle QPR.​

Answer 1

Answer:

1. Answer
2. AnswerGiven, Bisectors of
3. AnswerGiven, Bisectors of∠PQR
4. AnswerGiven, Bisectors of∠PQRand
5. AnswerGiven, Bisectors of∠PQRand∠PRS
6. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at point
7. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT
8. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.
10. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove:
11. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=
12. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=2
13. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=21
14. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=21∠QPR
15. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=21∠QPR.
17. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=21∠QPR.Proof,
19. AnswerGiven, Bisectors of∠PQRand∠PRSmeet at pointT.To prove: ∠QTR=21∠QPR.Proof,∠TRS=

∠TQR+

∠QTR

(Exterior angle of a triangle equals to the sum of the two interior angles.)

⇒

∠QTR=

∠TRS−

∠TQR

--- (i)

Also

∠SRP=

∠QPR+

∠PQR

2∠TRS= ∠QPR+ 2∠TQR

∠QPR=

2∠TRS−

2∠TQR

⇒

2

1

∠QPR=

∠TRS−

∠TQR

--- (ii)

Equating (i) and (ii),

∴

∠QTR=

2

1

∠QPR

[henceproved