Q. 85.0( 4 Votes )

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

Answer :

We have



Given: In ∆PQR,


QX bisects the Q & RX bisects the R.


XS QR & XT PQ.


To Prove: ∆XTQ ∆XSQ


& PX bisects P.


Proof: In ∆XSQ and ∆XTQ, we have


XSQ = 90°


XTQ = 90°


XSQ = XTQ …(i)


QX = QX [ This is the common side] …(ii)


XQS = TQX [ it is given that QX bisects Q] …(iii)


So, by (i), (ii) & (iii), we can say that


ΔXTQ ΔXSQ by AAS rule of congruency.


XS = XT [ corresponding parts of congruent triangles are always equal] …(iv)


Draw XW perpendicular to PR.


We have



Similarly, we can prove that Δ XSR is congruent to Δ XWR.


So, XS = XW …(v)


So, from (iv) and (v), we can say


XT = XW …(vi)


In ∆PXT and ∆PXW,


PTX = PWX [ PTX = PWX = 90°]


PX = PX [ they are common sides]


XT = XW [from (vi)]


BY R.H.S. both triangles are congruent,


That is, ∆PXT ∆PXW.


XPT = XPW [ corresponding parts of congruent triangles are always equal]


PX is bisector of P.


Hence, proved.


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