I see you've uploaded a geometry worksheet involving central angles and arc measures. Let me break down and solve each problem systematically:

1)

• Given a central angle of $100^\circ$, the arc measure is equal to the central angle.
• Answer: The arc measure is $100^\circ$.

2)

• Given one central angle of $120^\circ$, and assuming the entire circle is $360^\circ$, the unknown arc can be found by subtracting: $360^\circ - 120^\circ = 240^\circ$
• Answer: The arc measure is $240^\circ$.

3)

• Two central angles are given: $75^\circ$ and $140^\circ$.
• Find the remaining arc measure by subtracting the sum of the two angles from $360^\circ$: $360^\circ - (75^\circ + 140^\circ) = 360^\circ - 215^\circ = 145^\circ$
• Answer: The arc measure is $145^\circ$.

4)

• The measure of $\angle LHJ$ is asked, given adjacent angles $60^\circ$ and $150^\circ$.
• Since $\angle LHJ$ is the sum of the two given angles: $\angle LHJ = 60^\circ + 150^\circ = 210^\circ$
• Answer: $m\angle LHJ = 210^\circ$.

5)

• The central angles given are $40^\circ$ and $61^\circ$.
• To find $m\angle YTU$, subtract the given angles from $360^\circ$: $360^\circ - (40^\circ + 61^\circ) = 360^\circ - 101^\circ = 259^\circ$
• Answer: $m\angle YTU = 259^\circ$.

6)

• Central angles $91^\circ, 85^\circ,$ and $64^\circ$ are given.
• Find the measure of arc $UW$ by adding the central angles: $m\angle UW = 91^\circ + 85^\circ + 64^\circ = 240^\circ$
• Answer: $m\angle UW = 240^\circ$.

Would you like a detailed explanation of any step?

Additional Practice:

1. What is the total sum of central angles in any circle?
2. How do you calculate an unknown arc measure when given other arc measures?
3. What is the relationship between a diameter and a semicircle in terms of arc measure?
4. How would you approach solving for a missing central angle using algebra?
5. Can a single central angle ever be larger than 180 degrees? Why or why not?

Tip: Always remember that the sum of central angles in a circle equals $360^\circ$.