In Fig. 4, find the measure of arc ADC, if ∠OAB = 30° and ∠OCB = 50°.

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Solution

We have,

∠OAB = 30° and ∠OCB = 50°

In $∆$ AOB,

As, OA = OB (Radii)

So, ∠OAB = ∠OBA = 30° (Angles opposite to equal sides are equal)

Similarly, in $∆$ AOB,

As, OC = OB (Radii)

So, ∠OCB = ∠OBC = 50° (Angles opposite to equal sides are equal)

Therefore, ∠ABC = ∠OBA + ∠OBC = 30° + 50° = 80°

Now using the property of circles, the angle subtended by a chord at the center is twice the angle subtended by it at the remaining part of the circle.

∠AOC = 2∠ABC = 2 $\times$ 80° = 160°

Hence, the measure of arc ADC is 160°.

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