https://file.aiquickdraw.com/mathbot/file/1725884174617gbyghbls.jpg

To find the measure of \(\angle RUS\), we can use the fact that the angles around point \(U\) sum up to \(360^\circ\).

Given angles:
- \(\angle P U Q = 82^\circ\)
- \(\angle Q U R = 24^\circ\)
- \(\angle S U T = 51^\circ\)

Since \(\angle P U T\) is a straight angle (line), it equals \(180^\circ\). Therefore, we can calculate \(\angle RUS\) using the following steps:

1. Calculate \(\angle QUT\):
   \[
   \angle QUT = \angle P U T - \angle P U Q = 180^\circ - 82^\circ = 98^\circ
   \]
   
2. Calculate \(\angle RUS\):
   \[
   \angle RUS = \angle QUT - \angle QUR - \angle SUT = 98^\circ - 24^\circ - 51^\circ = 23^\circ
   \]

Thus, the measure of \(\angle RUS\) is \(23^\circ\).

### Final Answer:
The correct answer is **23°**.

Would you like more details or have any further questions? 

Here are five related questions to deepen your understanding:
1. How would you calculate \(\angle QUS\) if the measure of \(\angle RUS\) was not given?
2. What is the sum of the angles around a point, and why is it significant?
3. How can the concept of supplementary angles help solve problems involving straight angles?
4. How would the solution change if \(\angle SUT\) were different?
5. What other geometric properties can be used to solve for unknown angles?

**Tip:** Remember that the sum of angles around a point is always \(360^\circ\). This is a fundamental property used frequently in geometry problems.

Find the measure of ∠RUS from the image.

This problem asks to find the measure of ∠RUS using the sum of angles around point U. It uses key concepts from geometry including angle measures and the sum of angles around a point, suitable for Grades 7-9.

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