Question

Prove that If the bisector of vertical angles of a triangle bisects the base of the triangle,then the triangle is isosceles.

Answer 1

SoLuTiOn :

Let the triangle ABC in which AD is the bisector of angle A meeting BC in D such that BC = DC.

To Prove :

Triangle ABC is an isosceles Triangle

Construction :

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

Proof :

$\sf In \: \triangle \: EDC \: and \: ADB, \: we \: have :$

$\sf \bullet \: DC = BD \: \: (Given)$

$\sf \bullet \: DE = AD \: \: (Construction)$

$\sf \bullet \: \angle EDC = \angle ADB$

So,By SAS congruence criteria we have

$\sf \: \triangle \: EDC \cong \triangle \: ADB$

$\longrightarrow \sf \: EC = AB$

$\sf \: and, \: \angle CED = \angle BAD \: \: (c.p.c.t.)...i$

$\sf \: but, \angle CAD = \angle BAD \: (Given)$

$\therefore \: \sf \angle CED = \angle CAD$

$\longrightarrow \sf \: CE = AC \: (Side \: opposite \: to \: equal \: angle)$

$\longrightarrow \sf \: AB = AC \: (from \: eq.i )$

Hence,Triangle ABC is an isosceles Triangle

Answer 2

Step-by-step explanation:

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