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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