Question

In $ Fig. 10.36$ $ A, B$ and $ C$ are three points on a circle with centre $ O$ such that $ \angle BOC=30°$ and $ \angle AOB=60°$. If $ D$ is a point on the circle other than the arc $ \angle ABC$, find $ \angle ADC$.

Fig. 10.36

Answer 1

Find the given angle

We need to find $\angle ADC$

Given,

$\angle BOC=30°$ and $\angle AOB=60°$

So,

$$\begin{gathered} \Rightarrow \angle AOC=2\angle ADC\left[\because \mathrm{Angle}\mathrm{subtended}\mathrm{by}\mathrm{an}\mathrm{arc}\mathrm{is}\mathrm{double}\mathrm{the}\mathrm{angle}\mathrm{subtended}\mathrm{by}\mathrm{its}\mathrm{point}\mathrm{on}\mathrm{the}\mathrm{remaining}\mathrm{part}\mathrm{of}\mathrm{the}\mathrm{circle}\right] \\ \Rightarrow \angle AOB+\angle BOC=2\angle ADC \\ \Rightarrow 60°+30°=2\angle ADC \\ \Rightarrow 90°=2\angle ADC \\ \Rightarrow \angle ADC=45° \end{gathered}$$

Hence $\angle ADC=45°$