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Answer :

In the given figure, let us extend BA to P such that;

AP = AC.

Now join PC.

Given, \(\frac{BD}{CD} = \frac{AB}{AC} \)

\(\frac{ BD}{CD} = \frac{AP}{AC} \)

By using the converse of basic proportionality theorem, we get,

AD || PC

\(\angle\) BAD = \(\angle\) APC (Corresponding angles) ……………….. (i)

And, \(\angle\) DAC = \(\angle\) ACP (Alternate interior angles) …….… (ii)

By the new figure, we have;

AP = AC

\(\angle\) APC = \(\angle\) ACP ……………………. (iii)

On comparing equations (i), (ii), and (iii), we get,

\(\angle\) BAD = \(\angle\) APC

Therefore, AD is the bisector of the angle BAC.

Hence proved.

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