​Solve this:

Q6. In the adjoining figure, ABC is a triangle in which AD is the bisector of    A. If AD BC, show that  ABC is isosceles.


In ADB and ADCBAD = CAD  as, AD bisects A       AD = AD          commonADB = ADC  90° eachADB  ADC  ASAAB = AC    CPCTABC is an isosceles .

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Dear Chesna,
Here is the prove:
To Prove: ​△ABC is an isosceles triangle.
To prove the above statement, we have to prove, AB = AC as opposite sides of isosceles triangle are equal. 

In the adjoining figure, 
AD ⊥ BC.
ADC = 90​°
ADB = 90° .... (1)

Lets take consider Triangles ADB and ADC, 
In ​Triangles ADB and ADC,

∠BAD = ∠CAD {As ∠A is bisected}
AD = AD {Common side}
∠ADB = ​∠ADC (=90°) {Using[1]}

So, by the above mentioned factors, we can say that ADB ​≅ ADC by ASA congruency rule.

AB = AC {By C.P.C.T}

In an isosceles triangle the opposite sides should be equal. 
Thus, △ ABC is an isoceles triangle. 

Hope ithelps!
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