https://file.aiquickdraw.com/mathbot/file/1729413764892xm7hunum.jpg

Let's analyze the image and solve the given questions:

### **Question 2: Find the measure of arc \( R\hat{S}T \) in circle P**

We are provided with the following:
- \(\angle P\) between \( T \) and \( Q \) is given as \( 86^\circ \)
- The arc \( S\hat{T} \) is \( 145^\circ \).

The measure of the entire circle is \( 360^\circ \). To find the arc \( R\hat{S}T \), we need to consider the remaining section of the circle after excluding the arcs for \( S\hat{T} \) and \( T\hat{Q} \).

The missing arc, \( R\hat{S}T \), can be calculated as follows:
\[
360^\circ - (86^\circ + 145^\circ) = 360^\circ - 231^\circ = 129^\circ.
\]
Thus, the correct answer is:
\[
\boxed{129^\circ} \quad \text{(option b)}.
\]

---

### **Question 3: Find the measure of arc \( E\hat{G}F \)**

The details provided do not show a specific diagram or information related to \( EGF \), but we can infer the same steps as the previous problem. If additional information or the diagram is shared for this part, I'll help solve it.

Would you like more details or help with any part of this? Let me know!

### Relative Questions:
1. What is the relationship between central angles and arc measures in a circle?
2. How can you calculate the total of all arcs in a circle?
3. If one arc's measure is known, how do you find the remaining arcs in a circle?
4. How are minor and major arcs defined in a circle?
5. What happens when two central angles are adjacent? How do their arc measures relate?

### Tip:
In a circle, the sum of all central angles is always \( 360^\circ \), and each angle subtends an arc equal in measure to the angle itself.

Find the measure of arc RST in circle P and find the measure of arc EGF.

Learn how to calculate the arc measures in circle geometry problems. This example covers finding the measure of arc RST and arc EGF using basic circle theorems and properties of central angles. Suitable for students in Grades 8-10.

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