Question

PX bisects angle P. If True enter 1 else if False enter 0.

Answer 1

Given: $△PQR$, $QX$ and $RX$ are bisectors of $\angle Q$ and $\angle R$ and $XT\perp PQ$ and $XS\perp QR$
Construction: Draw $XM\perp PR$, Join PX
In $△XSR$ and $△XMR$
$\angle XRS=\angle XRM$ ($XS$ bisects $\angle R$)
$XR=XR$
$\angle XSR=\angle XMR={90}^{\circ }$
Thus, $△XSR\cong △XMR$ (ASA rule)
Hence, $XS=XM$ (BY CPCT)...(1)
Similarly, $△XQS\cong △XQT$
Hence, $XS=XT$ (By CPCT)....(2)
Hence, by (1) and (2)
$XM=XT$
Now, In $△PXT$ and $△PXM$
$XT=XM$ (Proved above)
$\angle XMP=\angle XTP$ (Each ${90}^{\circ }$)
$PX=PX$ (Common)
Thus, $△PXT\cong △PXM$ (SAS rule)
Hence, $\angle TPX=\angle MPX$ (by CPCT)
or $PX$ bisects $\angle P$