In the below figure (not to scale), $∠ A B C = 2 ∠ A C B$ Find the angle made by $¯ ¯¯¯¯¯¯ ¯ A C$ at the centre of the circle, if $∠ B A C = 60^{\circ} .$

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Solution

In $△ A B C$
$∠ A B C + ∠ B A C + ∠ A C B = 180^{\circ} ⟹ 3 ∠ A C B = 120^{\circ} ⟹ ∠ A C B = 40^{\circ}, ∠ A B C = 80^{\circ}$
The angle made by $A C$ at the centre will be $2$ times the angle made by $A C$ at any point on the circle
$⟹$ angle is $2 \times ∠ A B C = 160^{\circ}$

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In the given figure, O is the centre of the circle. If $∠$ ABD=35 $^{\circ}$ and $∠$ BAC=70 $^{\circ}$ , find $∠$ ACB.

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