Question

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

If $m\angle 3=t$, express the measures of the other numbered angles in terms of t.

Short answer

The values of the measure of the angles ∠4, ∠5 and ∠6 are $90°-t$, and$90°-t$ 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 perpendiculars.

Therefore, by using the definition of perpendicular lines, the measure of the angle$\angle BOE$ 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 t for$m\angle 3$ into the equation (1).

$$\begin{gathered} 90°=m\angle 4+m\angle 3 \\ 90°=m\angle 4+t \\ 90°-t=m\angle 4 \end{gathered}$$

Therefore, the measure of the angle ∠4 is $90°-t$.

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 $90°-t$for $m\angle 4$into the equation (2).

$$\begin{gathered} 90°=m\angle 5+m\angle 4 \\ 90°=m\angle 5+90°-t \\ 90°-90°+t=m\angle 5 \\ t=m\angle 5 \end{gathered}$$

Therefore, the measure of angle ∠5 is t.

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

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

Therefore, by using the definition of perpendicular lines, the measure of the angle ∠BOD is 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 t for m∠5into the equation (3).

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

Therefore, the measure of angle ∠6 is $90°-t$.