Question

In the following figure, ${}^{\prime }{P}^{\prime }$ is point equidistance from two lines ${}^{\prime }{l}^{\prime }$ and ${}^{\prime }{m}^{\prime }$ intersecting at point ${}^{\prime }{A}^{\prime }$. Show that the line $AP$ bisect the angle between them.

Answer 1

REF.Image

Given l and m are intersecting line, at A,

P is the point such that perpendicular from P to l and m are equal $\therefore PM=PN$

PM=PN

AP bisects $\angle MAN$

In $\Delta PMA$ & $\Delta PNA$

PM=PN [Since P is equidistance from l & m]

$\angle PMA=\angle PNA={90}^{\circ }$

PA=PA (Common)

$\therefore \Delta PMA\cong \Delta PNA$ [R.H.S criteria]

$\angle MAP=\angle NAP\left[CPCT\right]$

Thus, AP bisects the angle between l and m.