Question : In $\triangle A B C $, AB and AC are produced to points D and E, respectively. If the bisectors of $\angle C B D$ and $\angle B C E$ meet at the point O, and $\angle B O C=57^{\circ}$, then $\angle A$ is equal to:
Option 1: 93°
Option 2: 66°
Option 3: 114°
Option 4: 57°
Correct Answer: 66°
Solution : Given: $\angle BOC = 57^\circ$ ⇒ $\angle BOC = 90^\circ – \frac{\angle BAC}{2}$ ⇒ $\frac{\angle BAC}{2} = 90^\circ – 57^\circ$ ⇒ $\frac{\angle BAC}{2} = 33^\circ$ ⇒ $\angle BAC = 66^\circ$ Hence, the correct answer is $66^\circ$.
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Question : In $\triangle$ABC, $\angle$A = 66°. AB and AC are produced at points D and E, respectively. If the bisectors of $\angle$CBD and $\angle$BCE meet at the point O, then $\angle$BOC is equal to:
Question : In $\triangle A B C, \mathrm{BD} \perp \mathrm{AC}$ at $\mathrm{D}$. $\mathrm{E}$ is a point on $\mathrm{BC}$ such that $\angle B E A=x^{\circ}$. If $\angle E A C=46^{\circ}$ and $\angle E B D=60^{\circ}$, then the value of $x$ is:
Question : AB is a chord in a circle with centre O. AB is produced to C such that BC is equal to the radius of the circle. C is joined to O and produced to meet the circle at D. If $\angle \mathrm{ACD}=32^{\circ}$, then the measure of $\angle \mathrm{AOD}$ is _____.
Question : Internal bisectors of $\angle$ B and $\angle$ C of $\triangle$ ABC meet at O. If $\angle$ BAC = $80^{\circ}$, then the value of $\angle$ BOC is:
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