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In the figure, if ∠ACB = 35^o, then find the measure of ∠1.

Given: ∠ACB = 35^o

We know that the angle subtended by the chord at the centre is double the angle subtended by it at any point on the circle.

$\therefore \angle AOB=2\angle ACB$

$\Rightarrow \angle AOB=2\times {35}^o={70}^{\mathrm{o}}$

In △AOB,

OA = OB   [Radii of a circle]

We know that angles opposite the equal sides of a triangle are equal.

∴ ∠1 = ∠2

∠1 + ∠2 + ∠AOB = 180^o    [Angle sum property of a triangle]

⇒ ∠1 + ∠1 + 70^o = 180^o 

⇒ 2∠1 = 110^o 

⇒ ∠1 = 55^o