Question

In the adjoining figure,QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR. If XS perpendicular QR and XT perpendicular PQ prove that triangle XTQ is congruent to triangle XSQ and PX bisect angle P.

Answer 1

As the figure is not given, but I have tried my best to solve this question.
Solution: 
Angle XSQ = 90°
Angle XTQ = 90°
QX = QX (This is the common side)
Since QX bisects Q the angle is equally split between the triangles)
So, XQS = TQX
So, Δ XTQ is congruent to Δ XSQ
XS = XT (According to CPCT) ...(1)
Draw XW perpendicular to PR
Similarly, we can prove that Δ XSR is congruent to Δ XWR.
So, XS = XW ... (2)
So, from (1) and (2)
now in PXT and PXW, PTX = PWX, PX is common and XT = XW.
BY R.H.S. both triangles are congruent, XPT = XPW and PX bisects the angle P. Hence proved.

Answer 2

i] in Δ's XTQ and XSQ

 

  ∠ XSQ = 90°

  ∠ XTQ = 90°

  QX = QX (This is the common side)

  Since QX bisects Q the angle is equally split between the triangles)

  So, XQS = TQX

  So, Δ XTQ ≅ Δ XSQ

  XS = XT ( CPCT)  ____________(1]

  Similarly, we can prove that Δ XSR ≅ Δ XWR.

  So, XS = XW _______________(2)

  So, from (1) and (2)

  PXT and PXW, PTX = PWX, PX is common and XT = XW.

  BY RHS both triangles are congruent, XPT = XPW

  PX bisects the angle P.

  Hence proved.

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