Question

In the given figure, $△ABC$ is an isosceles triangle in which $AB=AC$ . If $AB$ and $AC$ are produced to $D$ and $E$ respectively such that $BD=CE$.

Is $BE=CD$ ?

• A. True
• B. False

Answer 1

The correct option is A True

Given:

$\left(1\right)$ $\Delta ABC$ is an isosceles triangle.

$⟹$ $AB=AC$.

$\left(2\right)$ $BC=CE$.

Since $AB=AC$,

$\therefore \angle ABC=\angle ACB$ [Since angles opposite to equal sides of a triangle are equal]

$\therefore {180}^o-\angle ABC={180}^o-\angle ACB$

$\angle DBC=\angle ECB$ ………..$\left(1\right)$

$[$ since $ABD$ and $ACE$ are straight lines$]$.

In triangles $DBC$ and $ECB$,

$\angle DBC=\angle ECB$ (from $\left(1\right)$)

$BD=CE$ (given)

$BC=BC$ (common side)

$\therefore \Delta DBC\sim \Delta ECB$ [by $SAS$ criteria].

So ,since triangles are congruent,

$\therefore DC=EB$.

Hence, $BE=CD$ is true.

Therefore, the statement is true.

Hence, option $A$ is correct.