Question

Given
\[\qquad m \angle AOC = 108^\circ\]
\[\qquad m \angle AOB = 3x + 4^\circ\]
\[\qquad m \angle BOC = 8x - 28^\circ\]
Find
\[m\angle AOB\]:
An obtuse angle AOC. The sum of angle AOB and angle BOC equals angle AOC.
\[O\]
\[A\]
\[C\]
\[B\]
\[{}^{\circ}\]

Answer 1

To solve for \( m \angle AOB \), we know that the sum of angles \( AOB \) and \( BOC \) equals angle \( AOC \). We can create the following equation based on the given information:

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

Substituting the expressions we have:

\[
(3x + 4^\circ) + (8x - 28^\circ) = 108^\circ
\]

Now, combine the terms on the left side:

\[
3x + 8x + 4^\circ - 28^\circ = 108^\circ
\]

This simplifies to:

\[
11x - 24^\circ = 108^\circ
\]

Next, we solve for \( x \) by adding \( 24^\circ \) to both sides:

\[
11x = 108^\circ + 24^\circ
\]

\[
11x = 132^\circ
\]

Now, divide by \( 11 \):

\[
x = \frac{132^\circ}{11} = 12^\circ
\]

Next, we can substitute \( x \) back into the expression for \( m \angle AOB \):

\[
m \angle AOB = 3x + 4^\circ = 3(12^\circ) + 4^\circ
\]

Calculating that gives:

\[
m \angle AOB = 36^\circ + 4^\circ = 40^\circ
\]

Now, let’s also find \( m \angle BOC \) to ensure everything checks out:

\[
m \angle BOC = 8x - 28^\circ = 8(12^\circ) - 28^\circ
\]

Calculating \( m \angle BOC \):

\[
m \angle BOC = 96^\circ - 28^\circ = 68^\circ
\]

Finally, we check that the angles sum up correctly:

\[
m \angle AOB + m \angle BOC = 40^\circ + 68^\circ = 108^\circ \quad \text{(which equals \( m \angle AOC \))}
\]

Thus, the result for \( m \angle AOB \) is:

\[
\boxed{40^\circ}
\]