Question

Side QR of ∆PQR is produced to a point S as shown in figure. The bisector of ⦟PQR & ⦟PRS meet at point T. Prove that ⦟QTR = ½ ⦟QPR.​

Answer 1

It is given that bisectors of ∠PQR and ∠PRS meet at point T.

⇒  ∠PRT=∠SRT          [ Since, T is an angle bisector of a ∠PRS ]

⇒  ∠PQT=∠TQR        [ Since, T is an angle bisector of a ∠PQR]

We know, the exterior angle of a triangle equals the sum of the two interior angles.

⇒  ∠TRS=∠TQR+∠QTR

⇒  ∠QTR=∠TRS−∠TQR            ---- ( 1 )

An exterior angle of a triangle equals the sum of the two interior angles.

⇒  ∠SRP=∠QPR+∠PQR

⇒  2∠TRS=∠QPR+2∠TQR           [ TR is a bisector of ∠SRP and QT is a bisector of ∠PQR ]

⇒  ∠QPR=2∠TRS−2∠TQR

⇒  ∠QPR=2(∠TRS−∠TQR)

⇒  1/2 ∠QPR=∠TRS−∠TQR  ----- ( 2 )

Equating ( 1 ) and ( 2 ) we get,

⇒  ∠QTR= 1/2∠QPR

         

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Answer 2

Answer:

Step-by-step explanation:

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