# In triangle PQR, PS is the bisector of angle QPR and PT perpendicular QR. Show that angle TPS = 1/2 (angle Q-angle R). Its urgent ...Answer Fast Please.....

given :PS is the bisector

PT is perpendicular

to prove : TPS = 1/2 (Q-R)

solution : angle qps = angle rps

angle qpt + angle tps = angle spr

in triangle prt,

angle ptq = 900

angle qpt + angle qpt = 900

angle q + 900 - angle qpt

in triangle prt,

angle ptr = 900

angle r + angle tpr +900

angle r + (angle tps + angle spr) - angle qpt

angle q - angle r = angle tps + (angle spr - angle qpt)

angle q - angle r = angle tps + angle tps

angle tps = 1/2 (angle q - angle r)

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Can U make it some more clear? Something goes wrong..

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Given; PS is bisector of angle QPR
PT is perpendicular to QR

To Prove; angle TPS = 1/2 (angle Q - angle R)

Solutio; We know that angle QPS = 1/2 angle P                                                                                                         (1)

angle Q + angle QPT = 90 degree
angle QPT = 90 degree -angle Q                                                                                                                   (2)

On putting equations (1) and (2)
angle TPS = angle QPS - angle QPT
= 1/2 angle P - ( 90 - angle Q )
= 1/2 angle P - 90 + angle Q
= 1/2 angle P - 1/2 (angle Q + angle P + angle R) + angle Q
= 1/2 angle P - 1/2 angle Q - 1/2 angle P - 1/2 angle R + angle Q
= -1/2 angle Q - 1/2 angle R + angle Q                             (+ve and -ve 1/2 angle  P got cancled )

by takig LCM in angle Q we get
-1/2 angle Q + 2/2 angle Q - 1/2 angle R
(-1/2 + 2/2) angle Q - 1/2 angle R

= 1/2 angle Q - 1/2 angle R
angle TPS = 1/2 ( angle Q -1/2 angle R )                                                                                (Hence proved)
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very long
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