Question

Given


\[\qquad \overline{OA}\perp\overline{OC}\]
\[\qquad m \angle BOC = 6x - 6^\circ\]
\[\qquad m \angle AOB = 5x + 8^\circ\]
Find
\[m\angle BOC\]:
A right angle AOC. The sum of angle AOB and angle BOC equals angle AOC.
\[O\]
\[A\]
\[C\]
\[B\]
\[{}^{\circ}\]

Answer 1

Given that \( \overline{OA} \perp \overline{OC} \), we know that \( m \angle AOC = 90^\circ \).

According to the problem, we have:

\[
m \angle BOC = 6x - 6^\circ
\]
\[
m \angle AOB = 5x + 8^\circ
\]

Since \( A \) and \( C \) create a right angle at point \( O \), the sum of angles \( AOB \) and \( BOC \) is equal to angle \( AOC \):

\[
m \angle AOB + m \angle BOC = m \angle AOC
\]

Substituting in the given expressions:

\[
(5x + 8^\circ) + (6x - 6^\circ) = 90^\circ
\]

Now, we can combine the terms:

\[
5x + 8 + 6x - 6 = 90
\]

This simplifies to:

\[
11x + 2 = 90
\]

Next, we solve for \( x \):

\[
11x = 90 - 2
\]
\[
11x = 88
\]
\[
x = \frac{88}{11} = 8
\]

Now substituting \( x \) back into the expression for \( m \angle BOC \):

\[
m \angle BOC = 6x - 6
\]
\[
m \angle BOC = 6(8) - 6
\]
\[
m \angle BOC = 48 - 6 = 42^\circ
\]

Thus, the measure of angle \( BOC \) is

\[
\boxed{42^\circ}
\]