In the figure the side qr of triangle pqr

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"Question 6 In the given figure, the side QR of ΔPQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR=∠QPR.

Class 9 - Math - Lines and Angles Page 108"

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Exterior Angle of a triangle:

If a side of a triangle is produced then the exterior angle so formed is equal to the sum of the two interior opposite angles.

______________________

Solution:

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.

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