Question

If vertex C, of $\angle PCQ=x$ lies on the circle. The arms of $\angle PCQ$ intersect the circle at A and B. If m(arc AB)= 120${}^{\circ }$, then find the value of $x$ .

Answer 1

By inscribed angle theorem, we know that the measure of an angle subtended by an arc at a point on the circle is half of the measure of the angle subtended by the arc at the centre.
Given, m(arc AB) = 120${}^{\circ }$
Then, $\angle PCQ=\angle ACB=x=\frac{120}{2}={60}^{\circ }$