Question

If the bisector of the vertical angle of triangle bisect the base, prove that the triangle is isosceles.

Answer 1

Construction

Produce $AD$ to $E$ such that $AD=DE$ Join $EC$

Proof : In $△ADB$ and $△EDC$

$BD=CD$ ( given)

$AD=ED$ ( by construction)

$\angle ADB=\angle EDC$ ( vertically opposite angles)

$\therefore △ADB\cong △EDC$ ( $SAS$ congruence)

$\therefore AB=ED-----\left(1\right)$ ( by $CPCT$)

and $\angle BAD=\angle CED$ ( by $CPCT$)

But $\angle BAD=\angle CAD$ (given)

$\therefore \angle CAD=\angle CED$

$\therefore AC=CE-------\left(2\right)$ ( sides opposite to equal angles of a triangle are equal.

From $\left(1\right)$ and $\left(2\right)$, $AB=AC$

$\therefore △ABC$ is isosceles.