In the diagram, $\overrightarrow{\mathrm{OB}}$bisects $\angle \mathbf{AOC}$and $\overleftrightarrow{EC}\mathit{\perp }\overleftrightarrow{OD}$.

Find the value of x.

$m\angle 5=2x$, $m\angle 3=x$.

The given diagram is:

As, $\overrightarrow{OB}$bisects the angle AOC, therefore the rayOB is the angle bisector of AOC.

Therefore, by using the angle bisector theorem it can be said that the measures of the angles∠4and∠5are equal.

That implies, $m\angle 4=m\angle 5$.

Therefore, the relation between the angles ∠4 and ∠5 is $m\angle 4=m\angle 5$.

From the given diagram it can be noticed that the angles∠3,∠4and∠5lies on the same straight-line EC.

Therefore, the sum of the angles ∠3, ∠4 and ∠5 is $180°$.

That implies $m\angle 3+m\angle 4+m\angle 5=180°$.

Therefore, the relation between the angles ∠3, ∠4 and ∠5 is $m\angle 3+m\angle 4+m\angle 5=180°$.

As $m\angle 3+m\angle 4+m\angle 5=180°$, $m\angle 4=m\angle 5$, $m\angle 5=2x$and $m\angle 3=x$.

Therefore, the value of x is given by :

$m\angle 3+m\angle 4+m\angle 5=180°$

$m\angle 3+m\angle 5+m\angle 5=180°$

$x+2x+2x=180°$

$5x=180°$

$x=\frac{180}{5}$

$x=36$

Therefore, the value of x is 36.