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In the diagram, the angles around point \( S \) must add up to 360°, as they form a complete circle around that point. The known angles are \( (9x - 5)^\circ \), \( (3x)^\circ \), and 163°, and there is an unknown angle \( \angle RSN \).

First, we can express the sum of all angles around point \( S \):

\[
(9x - 5) + 3x + 163 + \angle RSN = 360^\circ
\]

Since \( \angle RSN \) is a straight line (forming a linear pair with 163°), its value must be:

\[
\angle RSN = 180^\circ - 163^\circ = 17^\circ
\]

Now we can update the equation:

\[
(9x - 5) + 3x + 163 + 17 = 360^\circ
\]

Simplify the equation:

\[
12x + 175 = 360
\]

Now, solve for \( x \):

\[
12x = 360 - 175
\]
\[
12x = 185
\]
\[
x = \frac{185}{12} \approx 15.42
\]

Thus, \( x \approx 15.42^\circ \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you check if the sum of angles around a point is always 360°?
2. What is a linear pair of angles, and why is \( \angle RSN \) part of one?
3. What happens if one of the given angles is negative or 0?
4. How do you handle angles involving algebraic expressions?
5. How would the problem change if another unknown angle was introduced?

**Tip:** Always double-check that angles around a point add up to 360° for complete accuracy in geometry problems.

Find the value of x in the given diagram, where angles around point S include (9x - 5)°, (3x)°, and 163°, and the sum of all angles around point S must equal 360°.

Solve for x in a geometry problem involving angles around point S. The given angles are (9x - 5)°, (3x)°, and 163°, and the sum must equal 360°. Learn how to apply the sum of angles around a point and linear pair theorems to find the solution.

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