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Question 6
In the given figure, the side QR of ΔPQR is produced to a point S. If the bisector of PQR and PRS meet at point T, then prove that QTR=12QPR.

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Solution

Considering ΔQTR, TRS is an exterior angle.
QTR+TQR=TRS (exterior angle is equal to sum of opposite interior angles)
QTR=TRSTQR……………. (1)
For ΔPQR, PRS is an external angle.
QPR+PQR=PRS (exterior angle is equal to sum of opposite interior angles)
QPR+2TQR=2TRS (As QT and RT are angle bisectors, PQR=2TQR and PRS=2TRS)
QPR=2(TRSTQR)
QPR=2QTR [ By using equation (1)]
QTR=12QPR

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