If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.
Given:
The perpendiculars from any point within an angle on its arms are congruent
To do:
We have to prove that it lies on the bisector of that angle.
Solution:
Let a point $P$ lies in $\angle ABC$, $PL \perp BA$, $PM \perp BC$ and $PL = PM$.
$PB$ is joined.
![](/assets/questions/media/153848-52932-1632229361.png)
In $\triangle PLB$ and $\triangle PMB$
$PL = PM$
$PB = PB$ (Common)
Therefore, by RHS axiom,
$\triangle PLB \cong \triangle PMB$
This implies,
$\angle PBL = \angle PBM$ (CPCT)
Therefore,
$PB$ is the bisector of $\angle ABC$.
Hence proved.
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