Question

# In the adjoining figure ,QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR .If $$XT \perp QR$$ and $$XT \perp PQ$$ prove that :$$PX$$ bisects $$\angle P$$ Solution
Verified by Toppr

#### Given : A $$\Delta PQR$$ in which $$QX$$ is the bisectors of $$\angle Q$$ and $$RX$$ is the bisectors of $$\angle R$$$$XT \perp QR$$ and $$XT \perp PQ$$Construction : Draw $$XZ \cong PR$$ and join $$PX$$ProofIn $$\Delta XTQ$$ and $$\Delta XSQ$$ $$\angle TQX = \angle SQX$$ [$$QX$$ is the angle bisector of $$\angle Q$$]$$\angle XTQ = \angle XSQ=90^0$$ [Perpendicular to sides]$$QX = QX$$ [Common]By Angle - Angle - Side criterion of congruence,$$\Delta XTQ \cong \Delta XSQ$$ The corresponding parts of the congruent$$XT = XS$$ [ c.p.c.t] In $$\Delta XSR$$ and $$\Delta XZR$$$$\angle XSR=\angle XZR=90^0$$ ...[$$XS \bot QR$$ and $$\angle XSR=90^0$$]$$\angle SRX=\angle ZRX ...[$$RX$$is bisector of$$\angle R$$]$$RX=RX ...[Common]By Angle-Angle-Side criterion of congruence,$$\Delta XSR\cong \Delta XZR$$The corresponding parts of the congruent triangles are equal.$$\therefore XS=XZ$$ ...[C.P.C.T] ...(2)From (1) and (2)$$XT=XZ$$ ...(3)In $$\Delta XTP$$ and $$\Delta XZP$$$$\angle XTP=\angle XZP=90^0$$ ...[Given]$$XP=XP$$ ...[Common]$$XT=XZ$$ ...[From (3)]The corresponding parts of the congruent triangles are congruent$$\angle XPT = \angle XPZ$$So, $$PX$$ bisects $$\angle P$$ 4
Similar Questions
Q1

The diagonal PR of a quadrilateral PQRS bisects the angles P and R, then View Solution
Q2

In ∆PQR, PR > PQ, PA is the bisector of angle P and PM is perpendicular to QR. Prove that angle APM = 1/2 ( angle Q - angle R).

View Solution
Q3

In the given figure, P=30, Q=90, and PQ=11m. PX and QX are the respective angle bisectors of P and Q. If PX and QX are bisected perpendicularly by AY and BZ respectively, then ∠YXZ is: View Solution
Q4

In the given figure, P = 40°, Q= 90°, and PQ = 12 m. PX and QX are the respective angle bisectors of P and Q. If PX and QX are bisected perpendicularly by AY and BZ, respectively, then ∠YXZ is: View Solution
Q5

In the adjoining figure, QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR.
If XSQR and XTPQ ; then
i) ΔXTQΔXSQ
ii) PX bisects angle P.

both statement is ___ View Solution
Solve
Guides