Question: How are angles $\angle BOC$ and $\angle COD$ related? There are two correct answers.
Select the correct response(s):
They are adjacent angles
They are complementary
They are vertical angles
They are congruent angles
top-left: Diagram of lines and rays intersecting at $O$. A horizontal line runs through $O$ from $A$ on the left to $D$ on the right, a vertical ray goes up from $O$ to $B$, and a diagonal line runs from $E$ at bottom-left through $O$ to $C$ at upper-right. Angle $\angle BOC$ is the angle between $OB$ and $OC$ in the top-right region, and angle $\angle COD$ is the angle between $OC$ and $OD$ in the right region.
1. **Problem Statement:** Determine how angles $\angle BOC$ and $\angle COD$ are related given the described intersecting lines and rays at point $O$.
2. **Understanding the Angles:**
- $\angle BOC$ is formed by rays $OB$ and $OC$.
- $\angle COD$ is formed by rays $OC$ and $OD$.
3. **Check if they are adjacent angles:**
- Adjacent angles share a common side and vertex.
- Both angles share vertex $O$ and side $OC$.
- Therefore, $\angle BOC$ and $\angle COD$ are adjacent angles.
4. **Check if they are complementary:**
- Complementary angles sum to $90^\circ$.
- There is no information indicating their measures sum to $90^\circ$.
- So, they are not necessarily complementary.
5. **Check if they are vertical angles:**
- Vertical angles are opposite angles formed by two intersecting lines and are congruent.
- Here, $\angle BOC$ and $\angle COD$ share a ray $OC$, so they are not opposite angles.
- Hence, they are not vertical angles.
6. **Check if they are congruent angles:**
- Congruent angles have equal measure.
- Since $\angle BOC$ and $\angle COD$ are adjacent and share a ray, they are not necessarily congruent.
**Final answers:**
- They are adjacent angles.
- They are not complementary.
- They are not vertical angles.
- They are not congruent angles.
**However, the problem states there are two correct answers.**
Re-examining vertical angles:
- The diagram shows lines intersecting at $O$.
- $\angle BOC$ and $\angle AOD$ would be vertical angles.
- $\angle BOC$ and $\angle COD$ share a ray, so not vertical.
Re-examining congruent angles:
- Adjacent angles formed by intersecting lines can be supplementary but not necessarily congruent.
Therefore, the two correct answers are:
- They are adjacent angles.
- They are congruent angles (if the diagram implies $\angle BOC$ and $\angle COD$ are equal, e.g., if $OC$ bisects the angle between $OB$ and $OD$).
Without explicit measures, the safest two correct answers are:
**They are adjacent angles and they are congruent angles.**