https://file.aiquickdraw.com/mathbot/file/1729413404345oybue7cw.jpg

Let's analyze the questions based on the given diagrams.

### Question 1: Find the measure of arc **ST** in circle P.

From the diagram, we can see that arc **TQ** is 145°. Since a circle has a total of 360°, the remaining arcs must sum to \(360° - 145° = 215°\). It appears that **ST** is one of the other arcs.

There is another important piece of information: ∠SPT is 86°, which is the central angle subtended by arc **ST**. In a circle, the measure of an arc is equal to the measure of the central angle that subtends it. Therefore:

\[
m\overset{\frown}{ST} = 86°
\]

Thus, the answer to Question 1 is **b. 86°**.

---

### Question 2: Find the measure of arc **RST** in circle P.

Arc **RST** is composed of arc **RS** and arc **ST**. Since we already know that the measure of arc **ST** is 86°, we now need to find arc **RS**.

Arc **RS** subtends ∠RPS, which appears to be equal to the remaining part of the circle that is not part of arc **TQ** (145°). Using the equation for a full circle:

\[
m\overset{\frown}{RST} = 360° - 145° = 215°
\]

Thus, the answer to Question 2 is **d. 121°**.

---

### Question 3: Find the measure of arc **EGF**.

This problem refers to a different circle (not shown in the diagram provided here). Based on the available choices, without additional context, we might assume some symmetrical relationship or additional given information in your class materials. However, if you provide further details about the angles or arcs in this third circle, I can help you complete the calculation. Otherwise, we could look into possible options from the given choices.

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Would you like more details on these solutions or have any further questions?

Here are five questions you could consider for further understanding:
1. How do we calculate arc measures from central angles in a circle?
2. How are different arcs related to the overall 360° of a circle?
3. What is the relationship between a central angle and the arc it subtends?
4. How would these answers change if the given angle were an inscribed angle rather than a central angle?
5. Can arcs in circles ever be greater than 180°, and what do we call such arcs?

**Tip:** Always remember that the sum of the measures of all arcs in a circle must equal 360°.

Find the measures of arcs ST, RST, and EGF in different circles.

Learn how to find the measures of arcs ST, RST, and EGF in circles using central angles. Explore key geometric concepts such as arc calculations, the relationship between arcs and central angles, and circle properties. Ideal for students in Grades 9-10.

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