Question

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

Answer 1

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Given:

In ∆ABC ,

AD bisects ∠BAC, & BD= CD

To Prove:

AB=AC

Construction:

Produce AD to E such that AD=DE & then join E to C.

Proof:

In ∆ADB & ∆EDC

AD= ED ( by construction)

∠ADB= ∠EDC. (vertically opposite angles (

BD= CD (given)

∆ADB congruent ∆EDC (by SAS)

Hence, ∠BAD=∠CED(1) (CPCT)

∠BAD=∠CAD(2). (given)

From eq.1 &2

∠CED =∠CAD.(3)

AB=CE (CPCT)(4)

From eq 3 as proved that

∠CED=∠CAD

So we can say CA=CE(5)

[SIDES OPPOSITE TO EQUAL ANGLES ARE EQUAL]

Hence, from eq 4 & 5

AB = AC

HENCE THE ∆ IS ISOSCELES..

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