Question

the given figure, QT = QU and SR || TU. Prove that

(i) TU = UR.

(ii) RT bisects ∠URS.​

Answer 1

Given : QT = QU and SR || TU

To Find : Prove that  

(i) TU = UR.

(ii) RT bisects ∠URS.

Solution:

QT = QU

=> ∠QTU = ∠QUT    ( angles opposite to equal sides)

∠PQT = ∠QTU + ∠QUT  ( exterior angle of triangle = Sum of opposite interior angles)

=> 8y =  2∠QTU  = 2  ∠QUT

=> ∠QTU  = ∠QUT  = 4y

∠QUT = ∠UTR + ∠URT

=> 4y = 2y + ∠URT

=> ∠URT = 2y

=>  ∠UTR = ∠URT

Hence TU = UR

QED

Proved

SR || TU

∠QUT  = ∠URS   ( corresponding angles)

=> ∠URS = 4y

∠URS = ∠URT  + ∠SRT

=> 4y = 2y + ∠SRT

=> ∠SRT = 2y

=> ∠URT  =  ∠SRT = 2y

Hence RT bisects ∠URS.

QED

Hence proved

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