Question

In the given figure, $AD$ is perpendicular to $BC$ and $EF\parallel BC$, if $\angle EAB=\angle FAC$, show that triangles $ABD$ and $ACD$ are congruent.
Also, find the values of $x$ and $y$ if $AB=2x+3,AC=3y+1,BD=x$ and $DC=y+1$

Answer 1

$AD$ is perpendicular to $EF$
$\Rightarrow \angle EAD=\angle FAD={90}^{\circ }$
$\angle EAB=\angle FAC$ (given)
$\Rightarrow \angle EAD-\angle EAB=\angle FAD-\angle FAC$
$\Rightarrow \angle BAD=\angle CAD$
In $△ABD$ and $△ACD$
$\angle BAD=\angle CAD$ [proved above]
$\angle ADB=\angle ADC={90}^{\circ }$ [Given $AD$ is perpendicular on $BC$]
and $AD=AD$
$△ABD\cong △ACD$ [By ASA]
Hence proved.
$\angle ABD=\angle ACD\Rightarrow AB=AC$ and $BD=CD$ [By C.P.C.T]
$\Rightarrow 2x+3=3y+1$ and $x=y+1$
$\Rightarrow 2x-3y=-2$ and $x-y=1$
Substituting $2(1+y)-3y=-2$ Substituting $y=4$ in $x=1+y$
$x=1+y2+2y-3y=-2x=1+4$
$-y=-2-2x=5$
$-y=-4$