Question

In the alongside diagram, $ABCD$ is parallelogram. Prove that : $AB=2BC$

Answer 1

Given, $ABCD$ is a parallelogram

The bisectors of $\angle A,$ bisect $DC$ at $E$

$\therefore AB||CD$ [opposite sides of a parallelogram are equal]

$AB||CD$ and $AE$ is transversal

$\angle BAE=\angle AED$ [alternate angle]....(i)

$\angle DAE =\angle BAE [AE is the bisector of \angle A]....(ii)

From (i) and (ii),

$\angle AED=\angle DAE$

$\therefore$ in $△DAE$

$\angle AED=\angle DAE$

$\Rightarrow AD=DE$ [If two angles of a triangle are equal, then sides opposite to them are equal.]

$2AD=2DE=DE+DE=DE+EC$

$\Rightarrow 2AD=DC$

$AD=BC$ and $DC=AB\Rightarrow AB=2BC$