Let's go through the problems one by one based on the image provided:

9. Give two other names for PR:

In the diagram, segment $PR$ can be named in two other ways:

• $RP$ (because the order of the points can be reversed).
• Since point $P$ is on line $CD$ and it appears $PR$ is part of line $CD$, you can also use the name $PD$ or $DP$ (assuming that line includes those points).

10. Give two other names for $\angle CPR$:

The angle $\angle CPR$ has two alternate names:

• $\angle RPC$ (since the order of the points can be rearranged as long as the vertex is in the middle).
• $\angle DPC$ (assuming point $P$ lies on the same line and is relevant to the angle formed).

11. Name three collinear points:

Collinear points are points that lie on the same straight line. From the diagram, the following points appear to be collinear:

• Points $P$, $C$, and $D$ (on line $CD$).

13. Name three segments:

Three segments from the diagram can be:

• $PR$
• $CQ$
• $CP$

15. Name a pair of vertical angles:

Vertical angles are opposite angles formed by two intersecting lines. From the diagram:

• $\angle ACP$ and $\angle RCQ$ are vertical angles (since they are opposite to each other where lines intersect).

Would you like more details or have any questions on these? Here are some related questions to deepen your understanding:

1. What is the definition of opposite rays, and how could you identify them in the diagram?
2. Can you explain the difference between vertical angles and linear pairs?
3. How do you determine if points are collinear in a diagram without grid references?
4. What is the significance of naming a line segment in geometry? Does the order matter?
5. How would you explain the relationship between adjacent angles and linear pairs?

Tip: In geometry, the order of points matters when naming angles or segments, so be sure to keep the vertex in the middle for angles!