The image shows a diagram with four rays, $OA$, $OB$, $OC$, and $OD$, all emanating from a common point $O$. You are asked to find the measure of $\angle BOC$, given the following information:

• $m\angle AOC = 123^\circ$
• $m\angle AOB = 35^\circ$

Solution:

From the diagram, the angles $\angle AOB$, $\angle BOC$, and $\angle AOC$ are all around the same vertex $O$.

Since $\angle AOC$ encompasses $\angle AOB$ and $\angle BOC$, we can use the following relationship: $m\angle AOC = m\angle AOB + m\angle BOC$ Substituting the given values: $123^\circ = 35^\circ + m\angle BOC$ Solving for $m\angle BOC$: $m\angle BOC = 123^\circ - 35^\circ = 88^\circ$

Thus, the measure of $\angle BOC$ is $88^\circ$.

Do you want more details or have any questions?

Here are 5 related questions to expand on this concept:

1. How do you find the measure of an unknown angle when multiple angles share the same vertex?
2. What is the sum of the angles formed around a point?
3. How can you use the concept of supplementary angles in such diagrams?
4. Can you explain the difference between adjacent angles and vertical angles?
5. How would you find the measure of $\angle COD$ if it were part of this problem?

Tip: Always double-check the diagram to ensure all angles around a point are accounted for when solving such problems.