To solve this problem, we are tasked with finding $x$, the arc measure, and the labeled angle.

From the image:

1. The angle formed at $B$ is an external angle that intercepts the two arcs.
2. The measure of the external angle is given by the formula: $\text{External Angle} = \frac{1}{2} (\text{Difference of Intercepted Arcs}).$
3. Here, the external angle is represented as $16x + 4$, and the intercepted arcs are $38x + 2$ and the smaller arc.

Step 1: Set up the equation

The intercepted arcs are:

• The larger arc $38x + 2$,
• The smaller arc $0$ (as it's not labeled, we assume no contribution to the difference).

The external angle equation is: $16x + 4 = \frac{1}{2} (38x + 2 - 0).$

Step 2: Simplify and solve for $x$

$16x + 4 = \frac{1}{2} (38x + 2).$ Multiply through by 2 to eliminate the fraction: $2(16x + 4) = 38x + 2.$ Simplify: $32x + 8 = 38x + 2.$ Bring $x$-terms to one side: $32x - 38x = 2 - 8.$ $-6x = -6.$ Solve for $x$: $x = 1.$

Step 3: Find the arc and angle measures

1. Substitute $x = 1$ into the arc and angle expressions:

   • Larger arc: $38x + 2 = 38(1) + 2 = 40^\circ$,
   • External angle: $16x + 4 = 16(1) + 4 = 20^\circ$.

2. Since the smaller arc isn't labeled, the intercepted arc difference remains the larger arc ($40^\circ$).

Final Answer:

• $x = 1$,
• Larger arc $= 40^\circ$,
• External angle $= 20^\circ$.

Would you like a step-by-step explanation for any part?

Related Questions:

1. How is the formula for external angles derived?
2. What are other properties of angles and arcs in a circle?
3. Can you find the measure of the smaller arc in this context?
4. How does this change if the angle were inside the circle instead of external?
5. How do you use the intercepted arcs formula in real-world geometry problems?

Tip:

Always double-check whether arcs are major or minor when solving circle problems!