Question

If the bisector of the vertical angle of a triangle bisects the base, show that the triangle is isosceles.

Answer 1

Given $△ABC,$ $AD$ is a bisector of $\angle A$ which meets base $BC$ at $D$ such that $BD=DC$.

Produce $AD$ to meet $E$ such that $AD=ED$.

Now, in $△ABD$ and $△DEC$

$BD=DC$ ...... [Given]

$AD=DE$ ........ [By construction]

$\angle ADB=\angle EDC$ ..... [Vertically opposite angles]

$\therefore △ABD\cong △EDC$ [$\because SAS$ congruence ]

$⟹AB=EC$ and $\angle BAD=\angle DEC$ ..... [CPCT]

Also, $\angle BAD=\angle DAC$

$⟹\angle DAC=\angle DEC$

$⟹$ In $△ACE$, $\angle AEC=\angle CAE$

$⟹AC=CE$ ........ [Sides opposite to equal angles]

$⟹AB=AC$

Hence, $△ABC$ is isosceles.