Question

For exercises 3 and 4 you are given that $\overrightarrow{\mathbf{OB}}\mathbf{\perp }\mathbf{l}$and $\overrightarrow{\mathbf{OA}}\mathbf{\perp }\overrightarrow{\mathbf{OC}}$.

If $m\angle 3=37$, complete:

$m\angle 4=\underset{¯}{?}$ $m\angle 5=\underset{¯}{?}$ $m\angle 6=\underset{¯}{?}$

Short answer

The values of the measure of the angles $\angle 4$, $\angle 5$and$\angle 6$ are $53°$,$37°$ and$53°$ respectively.

Step-by-step solution

Step 1. Label the given diagram.

Label the given diagram as:

Step 2. Find the measure of the angle ∠4.

As $\overrightarrow{OB}\perp l$, therefore the lines OB and I are perpendicular.

Therefore, by using the definition of perpendicular lines, the measure of the angle∠BOE is 90°.

That implies, $m\angle BOE=90°$.

By using the angle addition postulate:

$m\angle BOE=m\angle 4+m\angle 3$

By using the relation $m\angle BOE=90°$, it can be obtained that:

$90°=m\angle 4+m\angle 3\left(1\right)$

Substitute 37° for$m\angle 3$into the equation (1).

role="math" localid="1646468400711" $$\begin{gathered} 90°=m\angle 4+m\angle 3 \\ 90°=m\angle 4+37° \\ 90°-37°=m\angle 4 \\ 53°=m\angle 4 \end{gathered}$$

Therefore, the measure of the angle ∠4 is 53°.

Step 3. Find the measure of the angle ∠5.

As $\overrightarrow{OA}\perp \overrightarrow{OC}$,therefore the lines OA and OC are perpendicular.

Therefore, by using the definition of perpendicular lines, the measure of the angle∠AOC is 90°.

That implies, $m\angle AOC=90°$.

By using the angle addition postulate:

$m\angle AOC=m\angle 5+m\angle 4$

By using the relation $m\angle AOC=90°$, it can be obtained that:

$90°=m\angle 5+m\angle 4\left(2\right)$

Substitute 53° for$m\angle 4$ into the equation (2).

role="math" localid="1646468920949" $$\begin{gathered} 90°=m\angle 5+m\angle 4 \\ 90°=m\angle 5+53° \\ 90°-53°=m\angle 5 \\ 37°=m\angle 5 \end{gathered}$$

Therefore, the measure of the angle ∠5 is 37°.

Step 4. Find the measure of the angle ∠6.

As$\overrightarrow{OB}\perp l$,therefore the lines OB and I are perpendicular.

Therefore, by using the definition of perpendicular lines, the measure of the angle∠BODis 90° .

That implies, $m\angle BOD=90°$.

By using the angle addition postulate:

$m\angle BOD=m\angle 5+m\angle 6$

By using the relation $m\angle BOD=90°$, it can be obtained that:

$90°=m\angle 5+m\angle 6\left(3\right)$

Substitute 37° for$m\angle 5$ into the equation (3).

$$\begin{gathered} 90°=m\angle 5+m\angle 6 \\ 90°=37°+m\angle 6 \\ 90°-37°=m\angle 6 \\ 53°=m\angle 6 \end{gathered}$$

Therefore, the measure of the angle ∠6 is 53°.