In the image, we are given an angle diagram with the following information:

• $m \angle AOB = 5w + 4$
• $m \angle DOE = 9w - 12$
• $\angle AOE$ is a straight angle, which means $m \angle AOE = 180^\circ$.
• We are tasked with finding $m \angle BOC$.

Step-by-step solution:

1. Since $\angle AOE$ is a straight angle, we know: $m \angle AOB + m \angle DOE = 180^\circ$ Substituting the given expressions for $m \angle AOB$ and $m \angle DOE$: $(5w + 4) + (9w - 12) = 180$

2. Simplify the equation: $5w + 9w + 4 - 12 = 180$ $14w - 8 = 180$

3. Solve for $w$: $14w = 188$ $w = \frac{188}{14} = 13.43$

4. Now, to find $m \angle BOC$, we observe that: $m \angle BOC = m \angle AOB$ Since $m \angle AOB = 5w + 4$, substitute $w = 13.43$ into the expression: $m \angle BOC = 5(13.43) + 4 = 67.15 + 4 = 71.15^\circ$

Thus, $m \angle BOC \approx 71.15^\circ$.

Do you need further details or have any questions?

Here are 5 related questions to consider:

1. What are complementary angles, and do they apply here?
2. How do you solve for angles in a triangle given their sum?
3. What is the difference between an acute and obtuse angle?
4. How do linear pairs of angles relate to this problem?
5. Can you solve for angles when only supplementary information is given?

Tip: Always verify angle sums when dealing with straight angles or triangles to check if the solution is consistent.