91Ó°ÊÓ

Label the given diagram as:

As $\stackrel{→}{OB}âŠ\yen l$, 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, $m∠BOE=90°$.

By using the angle addition postulate:

$m∠BOE=m∠4+m∠3$

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

$90°=m∠4+m∠3\left(1\right)$

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

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

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

As $\stackrel{→}{OA}âŠ\yen \stackrel{→}{OC}$,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, $m∠AOC=90°$.

By using the angle addition postulate:

$m∠AOC=m∠5+m∠4$

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

$90°=m∠5+m∠4\left(2\right)$

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

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

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

As$\stackrel{→}{OB}âŠ\yen l$,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, $m∠BOD=90°$.

By using the angle addition postulate:

$m∠BOD=m∠5+m∠6$

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

$90°=m∠5+m∠6\left(3\right)$

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

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

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