We are given the following information about circle XX:

• $m\angle YXZ = 45^\circ$
• $m\overset{\frown}{YZ} = (11x - 50)^\circ$

Step 1: Understand the relationship between the central angle and the arc

The measure of an inscribed angle $\angle YXZ$ is equal to half the measure of the intercepted arc $\overset{\frown}{YZ}$.

So, the formula relating the inscribed angle and the arc is:

$m\angle YXZ = \frac{1}{2} m\overset{\frown}{YZ}$

Step 2: Substitute the given values

We are given $m\angle YXZ = 45^\circ$ and $m\overset{\frown}{YZ} = (11x - 50)^\circ$. Substituting these into the equation, we get:

$45^\circ = \frac{1}{2}(11x - 50)^\circ$

Step 3: Solve for $x$

First, multiply both sides of the equation by 2 to eliminate the fraction:

$90^\circ = 11x - 50$

Now, add 50 to both sides:

$140^\circ = 11x$

Finally, divide by 11 to solve for $x$:

$x = \frac{140}{11} = 12.7$

Final Answer:

$x = 12.7$ (rounded to the nearest tenth).

---

Would you like further details on this solution or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the relationship between a central angle and its intercepted arc in a circle?
2. How do you find the measure of an angle if given the length of an arc?
3. What is the formula for an inscribed angle in terms of the intercepted arc?
4. How would the equation change if the angle were a central angle instead of an inscribed angle?
5. What are some practical applications of these angle-arc relationships in real-world geometry problems?

Tip: Always check if the angle is inscribed or central when working with circle theorems, as this changes the relationship with the arc!