Question

In the given figure, $ A$, $ B$, $ C$ and $ D$ are four points on a circle. $ AC$ and $ BD$ intersect at a point $ E$ such that $ \angle BEC=130°$ and $ \angle ECD=20°$. Find $ \angle BAC$

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Answer 1

Find the measure of $\angle BAC$

We need to find, $\angle BAC$

Given,

$\angle BEC=130°$

$\angle ECD=20°$

So,

In $∆EDC$

$$\begin{gathered} \Rightarrow \angle E=180°-130°\left[\mathrm{Linear}\mathrm{pair}\right] \\ \Rightarrow \angle E=50° \\ \Rightarrow \angle E+\angle C+\angle D=180°\left[\mathrm{Angle}\mathrm{sum}\mathrm{property}\mathrm{of}\mathrm{triangle}\right] \\ \Rightarrow 50°+20°+\angle D=180° \\ \Rightarrow \angle D=180°-70° \\ \Rightarrow \angle D=110° \end{gathered}$$

Since $\angle A$ and $\angle D$ lie in the same segment

$$\begin{gathered} \therefore \angle A=\angle D \\ \Rightarrow \angle A=110° \end{gathered}$$

Hence $\angle BAC=110°$