3.If the bisector of an angle of a triangle bisects the opposite side,

1.	3.If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.
Answer» 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 & ∆EDCAD= 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..

3.If the bisector of an angle of a triangle bisects the opposite side, prove that the triangle is isosceles.

Answer»

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 & ∆EDCAD= 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..