Question

In the diagram $\overrightarrow{OB}$bisects ∠AOC and $\overleftrightarrow{EC}\perp \overleftrightarrow{OD}$.

Find the value of X.

$m\angle 2=3x$,$m\angle 3=2x-4‎$

Short answer

The value of X is 23.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is :

Step 2. Write the relation between the angles ∠2 and ∠5.

From the given diagram it can be noticed that the angles∠2and∠5are the vertical angles.

Therefore, by using the theorem for vertical angles it can be said that the angles∠2and∠5have the same measure.

That implies, $m\angle 2=m\angle 5$.

Therefore, the relation between the angles ∠2 and ∠5 is $m\angle 2=m\angle 5$.

Step 3. Write the relation between the angles ∠4 and ∠5.

As, $\overrightarrow{OB}$ bisects the angle AOC, therefore the ray OB 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 $m\angle 4=m\angle 5$.

Step 4. Write the relation between the angles ∠2 and ∠4.

As, $m\angle 2=m\angle 5$and $m\angle 4=m\angle 5$.

Therefore,$m\angle 2=m\angle 4=3x$

Therefore, the relation between the angles ∠2 and ∠4 is $m\angle 2=m\angle 4=3x$.

Step 5. Write the relation between the angles ∠2, ∠3 and ∠4.

From the given diagram it can be noticed that the angles ∠2, ∠3 and ∠4 lies on the same straight line $BZ$.

Therefore, the sum of the angles ∠2, ∠3 and ∠4 is $180°$.

That implies $m\angle 2+m\angle 3+m\angle 4=180°$.

Step 6. Find the value of x.

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

Therefore, the value of X is given by :

$m\angle 2+m\angle 3+m\angle 4=180°$

$3x+2x-4+3x=180°$

$8x-4=180°$

role="math" localid="1646586059432" $8x=180°+4$

$8x=184°$

$x=\frac{184}{8}$

$x=23$

Therefore, the value of X is23.