Question

In the adjoining figure, three coplanar lines AB, CD and EF intersect at a point O. Find the value of x. Also, find ∠AOD, ∠COE and ∠AOE.

Answer 1

We know that if two lines intersect, then the vertically-opposite angles are equal.
$$\begin{gathered} \therefore \angle DOF=\angle COE=5x° \\ \angle AOD=\angle BOC=2x°\text{and} \\ \angle AOE=\angle BOF=3x° \end{gathered}$$

Since, AOB is a straight line, we have:
$$\begin{gathered} \angle AOE+\angle COE+\angle BOC=180° \\ \Rightarrow 3x+5x+2x=180° \\ \Rightarrow 10x=180° \\ \Rightarrow x=18° \end{gathered}$$

Therefore,
$$\begin{gathered} \angle AOD=2\times 18°=36° \\ \angle COE=5\times 18°=90° \\ \angle AOE=3\times 18°=54° \end{gathered}$$