We are given a triangle $PQR$ with the following information:

1. $\angle P = 3x + 50$
2. $\angle Q = 2x + 9$
3. $\angle R = 4y + 5$
4. When side $QR$ is extended through point $R$ to point $S$, the measure of $\angle PRS = x + y$.

We need to determine the value of $x + y$.

Step 1: Sum of angles in triangle $PQR$

The sum of the interior angles of any triangle is $180^\circ$. Therefore, we can write the equation for the sum of the angles in triangle $PQR$ as:

$(3x + 50) + (2x + 9) + (4y + 5) = 180$

Simplify the left-hand side:

$5x + 4y + 64 = 180$

Now, subtract 64 from both sides:

$5x + 4y = 116 \quad \text{(Equation 1)}$

Step 2: Exterior angle relationship

The exterior angle $\angle PRS$ is equal to the sum of the two non-adjacent interior angles of the triangle. Therefore, we have:

$\angle PRS = \angle P + \angle Q$

Substitute the given expressions for $\angle P$ and $\angle Q$:

$x + y = (3x + 50) + (2x + 9)$

Simplify the right-hand side:

$x + y = 5x + 59 \quad \text{(Equation 2)}$

Step 3: Solving the system of equations

Now, we have two equations:

1. $5x + 4y = 116$
2. $x + y = 5x + 59$

Solve Equation 2 for $y$:

$y = 5x + 59 - x$ $y = 4x + 59$

Step 4: Substituting into Equation 1

Substitute $y = 4x + 59$ into Equation 1:

$5x + 4(4x + 59) = 116$

Simplify:

$5x + 16x + 236 = 116$ $21x + 236 = 116$

Subtract 236 from both sides:

$21x = -120$

Solve for $x$:

$x = -\frac{120}{21} = -\frac{40}{7}$

Step 5: Solving for $y$

Substitute $x = -\frac{40}{7}$ into $y = 4x + 59$:

$y = 4\left(-\frac{40}{7}\right) + 59$ $y = -\frac{160}{7} + 59$ $y = -\frac{160}{7} + \frac{413}{7} = \frac{253}{7}$

Step 6: Finding $x + y$

Now, calculate $x + y$:

$x + y = -\frac{40}{7} + \frac{253}{7} = \frac{213}{7}$

Thus, the value of $x + y$ is $\frac{213}{7}$ or approximately 30.43.

Let me know if you'd like further clarification or have any questions!

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Here are 5 related questions to deepen your understanding:

1. How do you apply the exterior angle theorem in different types of triangles?
2. What is the importance of understanding angle relationships in polygons?
3. How would the solution change if the angles were given in terms of other variables?
4. Can you solve a triangle with only two angles given?
5. How does the sum of angles change in non-Euclidean geometry?

Tip: Always double-check your equation simplifications to ensure the accuracy of your solution.