Question

Q.No.9. The side QR of a triangle PQR is produced to a point S. If the bisector of
ZPQR and Z PRS meet at point T, QPR (See given Figure), then prove that;
Z QTR = ZQPR
R​

Answer 1

In ΔQTR,

∠ TRS = ∠ TQR + ∠ QTR             Exterior angle theorem in a triangle

∠ QTR = ∠ TRS - ∠ TQR ......(I)

 

Also in ΔQPR,

∠ SRP = ∠ QPR + ∠ PQR  

 

2∠ TRS = ∠ QPR + 2∠ TQR       ∠TRS and ∠TQR are the bisectors of ∠SRP and ∠PQR respectively

 

∠ QPR = 2 ∠ TRS - 2 ∠ TQR

∠ TRS - 2 ∠ TQR = 1/2 ∠ QPR .....(II)

From (I) and (II), we get

∠ QTR = 1/2 ∠ QPR  

 

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Answer 2

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Class 7

>>Maths

>>The Triangle and Its Properties

>>Exterior Angle of a Triangle

>>In Figure, the side QR of PQR is produc

Question



In Figure, the side QR of △PQR is produced to a point S. If the bisectors of ∠PQR and ∠PRS meet at point T, then prove that ∠QTR=21∠QPR.





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Solution



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Given, Bisectors of ∠PQRand ∠PRS meet at point T.

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 

⇒21∠QPR=∠TRS−∠TQR --- (ii)

Equating (i) and (ii),

∴∠QTR=21∠QPR   [henceproved]