In the given figure, $BE$ and $CF$ are two equal altitudes of $△ABC$. Show that $△ABE\cong △ACF$,

 

 In the given figure, $BE$ and $CF$ are two equal altitudes of $△ABC$. Show that

$AB=AC.$

$△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC.$ If $AD$ is extended to intersect $BC$ at $E$, show that $△ABD\cong △ACD$

$△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC.$ If $AD$ is extended to intersect $BC$ at $E$, show that $△ABE\cong △ACE$.

$△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC.$ If $AD$ is extended to intersect $BC$ at $E$, show that $AE$ bisects $\angle A$ as well as $\angle D$.

$△ABC$ and $△DBC$ are two isosceles triangles on the same base $BC$ and vertices $A$ and $D$ are on the same side of $BC.$ If $AD$ is extended to intersect $BC$ at $E$, show that $AE$ is the perpendicular bisector of $BC$.

In the given figure, if $x=y$ and $\mathrm{AB}=\mathrm{CB}$, then prove that $\mathrm{AE}=\mathrm{CD}$.

In the given figure, line $L$ is the bisector of an angle $\angle A$ and $B$ is any point on $L$. If $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$, show that

$△APB\cong △AQB$

In the given figure, line $L$ is the bisector of an angle $\angle A$ and $B$ is any point on $L$. If $BP$ and $BQ$ are perpendiculars from $B$ to the arms of $\angle A$, show that

$BP=BQ$, i.e., $B$ is equidistant from the arms of $\angle A$.