We are given a circle with center $B$ and $m\overset{\frown}{DC} = 109^\circ$. We need to find $x$ in the equation $m\angle DBC = (9x + 24)^\circ$.

Step-by-step solution:

1. Recognize that $m\angle DBC$ is an inscribed angle: An inscribed angle is half of the measure of the intercepted arc. Here, the intercepted arc is $\overset{\frown}{DC}$, which is given as $109^\circ$.

   So, the measure of the inscribed angle $m\angle DBC$ can be calculated using the formula: $m\angle DBC = \frac{1}{2} \times m\overset{\frown}{DC}$ Substituting the value of $m\overset{\frown}{DC} = 109^\circ$: $m\angle DBC = \frac{1}{2} \times 109^\circ = 54.5^\circ$

2. Set up the equation: We are given that $m\angle DBC = (9x + 24)^\circ$, so we can set up the equation: $9x + 24 = 54.5$

3. Solve for $x$:

   • Subtract 24 from both sides: $9x = 54.5 - 24$ $9x = 30.5$
   • Divide by 9: $x = \frac{30.5}{9} \approx 3.39$

Thus, $x \approx 3.4$ (rounded to the nearest tenth).

Would you like more details, or do you have any questions?

Here are five related questions for further practice:

1. If the intercepted arc $\overset{\frown}{DC}$ were $120^\circ$, what would be the value of $x$?
2. How would the solution change if $m\overset{\frown}{DC} = 150^\circ$?
3. What is the relationship between an inscribed angle and the arc it intercepts?
4. If $m\angle DBC$ were given as $10x + 5$, how would you solve for $x$?
5. How can you generalize the formula for finding an inscribed angle given any arc?

Tip: When dealing with circles, always remember that the inscribed angle is half the measure of its intercepted arc.