Question

If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.  

Answer 1

Given that, if perpendicular from any point within, an angle on its arms is congruent, prove that it lies on the bisector of that angle
Now,
Let us consider an angle ABC and let BP be one of the arm within the angle  

Draw perpendicular PN and PM on the arms BC and BA such that they meet BC and BA in N and M respectively. 

Now, in ΔBPM and ΔBPN
We have ∠BMP= BNP = 90°     [given] 

BP=BP                                  [Common side] 

And MP=NP                            [given]   

So, by RHS congruence criterion, we have 

ΔBPM≅ΔBPN  

Now, 

∠MBP=∠NBP        [ Corresponding parts of congruent                                            triangles are equal] 

⇒ BP is the angular bisector of ∠ABC  

∴ Hence proved