PQRS is a parallelogram,T is the midpoint of PQ ,ST bisects angle S Prove that - a) QR = QT b) RT bisects Angle Rc)angle STR = 90o Share with your friends Share 21 Varun.Rawat answered this We have,PQRS as the given parallelogram, then PQ∥RS and PS∥QR.Since PQ∥RS and TS is a transversal, then∠PTS = ∠TSR Alternate interior angles .....1Now, TS bisects ∠S, then∠PST = ∠TSR ........2From 1 and 2, we get∠PTS = ∠PSTIn ∆PTS,∠PTS = ∠PST Proved above⇒PS = PT Sides opposite to equal angles are equal .........3But PS = QR opposite sides of parallelogram are equal ........4PT = TQ As T is the mid point of PQ ..........5From 3,4 and 5, we getQR = TQIn ∆TQR,QR = TQ Proved above⇒∠QTR = ∠QRT angles opposite to equal sides are equal .........6Since PQ∥RS and TR is a transversal, then∠TRS = ∠QTR Alternate interior angles .........7From 6 and 7, we get∠QRT = ∠TRS⇒TR bisects ∠R.Now, ∠R + ∠S = 180° Adjacent angles of ∥gm are supplementary⇒12∠R + 12∠S =90°⇒∠TRS + ∠TSR = 90° .......8In ∆TSR,∠TRS + ∠TSR + ∠RTS = 180° Angle sum property⇒90° + ∠RTS = 180°⇒∠RTS = 90° 7 View Full Answer
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