Question : The exterior angles obtained on producing the base $BC$ of a $\triangle ABC$ in both ways are $120^{\circ}$ and $105^{\circ}$, then the vertical $\angle A$ of the triangle is:
Option 1: $36^{\circ}$
Option 2: $40^{\circ}$
Option 3: $45^{\circ}$
Option 4: $55^{\circ}$
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Correct Answer: $45^{\circ}$
Solution :
Exterior angles obtained on producing the base $BC$ of a $\triangle ABC$ in both ways are $120^{\circ}$ and $105^{\circ}$.
⇒ $\angle ABC = 180^{\circ} -120^{\circ}=60^{\circ}$
⇒ $ \angle ACB= 180^{\circ}-105^{\circ}=75^{\circ}$
In $\triangle ABC$,
$\angle ABC+\angle ACB+\angle BAC=180^{\circ}$
⇒ $\angle BAC=180^{\circ}-60^{\circ}-75^{\circ}=45^{\circ}$
Hence, the correct answer is $45^{\circ}$.
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