Let's go through each question based on the diagrams and provided values.

For the first circle (with diameter $PQ$):

1. 10. $m\angle QOR$:

   • Since $PQ$ is the diameter, angle $QOR$ forms a central angle of $75^\circ$ as marked in the figure.
   • Answer: $m\angle QOR = 75^\circ$.

2. 11. $m\overarc{PR}$:

   • Arc $PR$ corresponds to the central angle $QOR = 75^\circ$.
   • Answer: $m\overarc{PR} = 75^\circ$.

3. 12. $m\overarc{QR}$:

   • Since $PQ$ is the diameter, the circle is divided into two semicircles, meaning $m\overarc{QR}$ is the remaining arc.
   • To find $m\overarc{QR}$, subtract $75^\circ$ from $180^\circ$.
   • Answer: $m\overarc{QR} = 180^\circ - 75^\circ = 105^\circ$.

4. 13. $m\overarc{PSQ}$:

   • Arc $PSQ$ represents the entire circle minus arc $PR$.
   • Answer: $m\overarc{PSQ} = 360^\circ - 75^\circ = 285^\circ$.

For the second circle (with diameter $FD$):

1. 14. $m\angle COF$:

   • $COF$ forms a central angle of $45^\circ$ as marked in the figure.
   • Answer: $m\angle COF = 45^\circ$.

2. 15. $m\overarc{FC}$:

   • Arc $FC$ corresponds to the central angle $COF = 45^\circ$.
   • Answer: $m\overarc{FC} = 45^\circ$.

3. 16. $m\overarc{CDE}$:

   • To find $m\overarc{CDE}$, add the central angles corresponding to arcs $CD$ and $DE$: $118^\circ + 75^\circ = 193^\circ$.
   • Answer: $m\overarc{CDE} = 193^\circ$.

4. 17. $m\overarc{CFE}$:

   • Arc $CFE$ represents the entire circle minus arc $FD$.
   • Since $FD$ is a diameter, it subtends a semicircle of $180^\circ$.
   • Answer: $m\overarc{CFE} = 360^\circ - 180^\circ = 180^\circ$.

Would you like further explanations on any of these steps?

Additional Related Questions

1. How can we calculate the measure of an arc given its central angle?
2. Why does a diameter divide a circle into two equal arcs?
3. How would you find the measure of a minor arc versus a major arc?
4. What properties of a circle help in calculating angle measures?
5. How do central angles relate to the arcs they intercept?

Tip

Remember, a central angle in a circle always equals the measure of the arc it intercepts.