Question
In the given figure$; AE$ bisects exterior angle $CAD$ and $AE$ is parallel to $BC.$
Prove that $: A B=A C$.
Prove that $: A B=A C$.
✓
Answer
Since $AE || BC$ and $DAB$ is the transversal.
$\therefore \angle DAE = \angle ABC = \angle B .......[$Corresponding angles$]$
Since $AE || BC$ and $AC$ are the transversals.
$\therefore \angle CAE = \angle ACB = \angle C ......[$Alternate angles$]$
But $AE$ bisects $\angle CAD,$
$\therefore \angle DAE = \angle CAE$
$\Rightarrow \angle B =\angle C$
$\Rightarrow AB = AC .......[$Sides opposite to equal angles are equal$]$
$\therefore \angle DAE = \angle ABC = \angle B .......[$Corresponding angles$]$
Since $AE || BC$ and $AC$ are the transversals.
$\therefore \angle CAE = \angle ACB = \angle C ......[$Alternate angles$]$
But $AE$ bisects $\angle CAD,$
$\therefore \angle DAE = \angle CAE$
$\Rightarrow \angle B =\angle C$
$\Rightarrow AB = AC .......[$Sides opposite to equal angles are equal$]$
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