Question

In an isosceles triangle $ABC$ with $AB=AC,D$ and $E$ are points on $BC$ such that $BE=CD$ Show that $AD=AE$

Answer 1

Given $ABC$ is an isosceles triangle with $AB=AC$
$D$ and $E$ are the point on $BC$ such that $BE=CD$

Given $AB=AC$

$\therefore \angle ABD=\angle ACE$...........(1)
[opposite angle of sides of a triangle]

Given $BE=CD$

Then $BE-DE=CD-DE$

$\Rightarrow BD=CE$.....................(2)

In $\Delta ABD$ and $\Delta ACE$

$\angle ABD=\angle ACE$ [From 1]

$BD=CE$ [From 2]

$AB=AC$ [Given]

$\therefore \Delta ABD\cong \Delta ACE$ by SAS congruency

By CPCT,

$AD=AE$ Proved