In fig. A,B and C are three points on a circle with centre O such that $\angle BOC={{30}^{\circ }}$ and $\angle AOB={{60}^{\circ }}$ . If D is appointed on the circle other than the arc $\triangle ABC$. Find $\angle ADC$ .

Hint:
Here we need to calculate the value of $\angle ADC$. For that, we will first calculate the angle subtended by $arcABC$ at the centre of the circle. To calculate that angle we will equate it to the sum of the two angles at the centre of a circle as their values are given in the question. Then we will have the property of an arc which states that the angle subtended by an arc at the centre of a circle is double the value of angle subtended by the same arc at any other point on the circumference of a circle. After equating the angle as stated we will get the value of the required angle.

Complete step by step solution:
It is given in the question that:
$\angle AOB={{60}^{\circ }}$ and $\angle BOC={{30}^{\circ }}$
The sum of $\angle AOB$ and $\angle BOC$is equal to $\angle AOC$
Which is the angle subtended by arc ABC at the centre.
Let’s calculate the value of $\angle AOC$
$\angle AOC=\angle AOB+\angle BOC$
Now, putting values of $\angle AOB$ and $\angle BOC$;
$\angle AOC={{60}^{\circ }}+{{30}^{\circ }}$
Now, adding terms on right hand side;
$\angle AOC={{90}^{\circ }}$………………$\left( 1 \right)$
So the angle subtended by an arc ABC at the centre of a circle is ${{90}^{\circ }}$
Now, using the property of an arc which states that the angle subtended by an arc at the centre of a circle is double the value of angle subtended by the same arc at any other point on the circumference of a circle, we get;
$\angle AOC=2\angle ADC$
Now, putting value of $\angle AOC$ as calculated in equation $\left( 1 \right)$;
${{90}^{\circ }}=2\angle ADC$
On dividing ${{90}^{\circ }}$ by 2 we get;
$\dfrac{{{90}^{\circ }}}{2}=\angle ADC$
Therefore,
$\angle ADC={{45}^{\circ }}$

So the angle subtended by an $arcABC$ at point $D$ is ${{45}^{\circ }}$.

Note:
We have used the property of an arc here which is defined as the portion of the circumference of the circle. Circumference itself can be considered as an arc which is called as full circle arc length.
There are two types of arc:-
Minor arc- An arc is called a minor arc if it subtends angle less than ${{180}^{\circ }}$ and its length is basically less than half of the circumference of a circle.
Major arc- An arc is called a minor arc if it subtends angle more than ${{180}^{\circ }}$ and its length is basically more than half of the circumference of a circle.