Question

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

Find the value of $x$.

localid="1649243329006" $m\angle 1=x-8$, width="99">m∠2=2x+5,$m\angle 4=3x-26$

Short answer

The value of $x$is 31.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Write the relation between the angles ∠1 and ∠2.

As, $\overleftrightarrow{EC}\perp \overleftrightarrow{OD}$, therefore the lines $EC$and$OD$ are perpendicular lines.

Therefore, by using the definition of perpendicular lines it can be said that the measures of the angle $\angle DOE$is $90°$.

That implies, $m\angle DOE=90°$.

Now, by using the angle addition postulate it can be said that $m\angle DOE=m\angle 1+m\angle 2$.

Therefore, the relation between the angles $\angle 1$and $\angle 2$is:

$\begin{array}{c}m\angle DOE=m\angle 1+m\angle 2\\ 90=m\angle 1+m\angle 2\end{array}$

- Find the value of x.

As, $m\angle 1+m\angle 2=90$, $m\angle 1=x-8$and $m\angle 2=2x+5$.

Therefore, the value of $x$is given by:

localid="1649244885505" $$\begin{gathered} m\angle 1+m\angle 2=90 \\ x-8+2x+5=90 \\ 3x-3=90 \\ 3x=90+3 \\ 3x=93 \\ x=\frac{93}{3} \\ x=31 \end{gathered}$$

Therefore, the value of$x$is 31.