Question

In the given figure, the side QR of triangle PQR is producer to a point S. If the bisector of angle PQR and angle PRS meet at point T, then prove that angle QTR is equal to half angle QPr in the given figure, the snake you are of triangle PQRS produce to a point S. If the bisector of angle PQR and angle PRS meet at point T, then prove that angle QTR is equal to half angle QPR

Answer 1

Given,Bisectors of ∠PQR & ∠PRS meet at point T.

To prove,

∠QTR = 1/2∠QPR.

Proof,∠TRS = ∠TQR +∠QTR

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

⇒∠QTR=∠TRS–∠TQR — (i)

∠SRP = ∠QPR + ∠PQR

⇒ 2∠TRS = ∠QPR + 2∠TQR

[ TR is a bisector of ∠SRP & QT is a bisector of ∠PQR]

⇒∠QPR= 2∠TRS – 2∠TQR

⇒∠QPR= 2(∠TRS – ∠TQR)

⇒ 1/2∠QPR =  ∠TRS – ∠TQR — (ii)

Equating (i) and (ii

∠QTR= 1/2∠QPR

Hence proved.

Answer 2

Hello mate ☺

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Solution:

∠PQT=∠TQR               (Given)

∠PRT=∠TRS               (Given)

To Prove:  ∠QTR=1/2(∠QPR)

∠PRS=∠QPR+∠PQR

(If a side of a triangle is produced, then the exterior angle is equal to the sum of two interior opposite angles.)

⇒∠QPR=∠PRS−∠PQR

⇒∠QPR=2∠TRS−2∠TQR

⇒∠QPR=2(∠TRS−∠TQR)

=2(∠TQR+∠QTR−∠TQR)                          (∠TRS=∠TQR+∠QTR)

(If a side of a triangle is produced, then the exterior angle is equal to the sum of two interior opposite angles.)

⇒∠QPR=2(∠QTR)

⇒∠QTR=1/2(∠QPR)

Hence Proved

I hope, this will help you.☺

Thank you______❤

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