Question : In $\triangle {ABC}$, O is the incentre and $\angle {BOC}=135^{\circ}$. The measure of $\angle {BAC}$ is:
Option 1: 90º
Option 2: 55º
Option 3: 80º
Option 4: 45º
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Correct Answer: 90º
Solution :
In a triangle, the angle between the incenter and any two sides is always $90^{\circ}$ plus half of the included angle. This is a property of the incenter of a triangle.
Given that $\angle {BOC}=135^{\circ}$, it means that the included angle at A (i.e., $\angle {BAC}$) is twice the difference of $\angle {BOC}$ and $90^{\circ}$.
$\angle {BAC} = 2 \times (\angle {BOC} - 90^{\circ}) = 2 \times (135^{\circ} - 90^{\circ}) = 90^{\circ}$
Hence, the correct answer is 90º.
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