Question

find the value of x, where $x^{circ}=moverarc{bd}$.

Explanation:

Step1: Recall secant angle theorem

The measure of an angle formed by two secants outside a circle is half the difference of the measures of the intercepted arcs. The formula is:
$$m\angle C = \frac{1}{2}(m\overset{\frown}{EA} - m\overset{\frown}{BD})$$

Step2: Substitute known values

Plug $m\angle C=31.8^\circ$, $m\overset{\frown}{EA}=111^\circ$, $m\overset{\frown}{BD}=x^\circ$ into the formula:
$$31.8 = \frac{1}{2}(111 - x)$$

Step3: Solve for x (isolate term)

Multiply both sides by 2 to eliminate the fraction:
$$31.8 \times 2 = 111 - x$$
$$63.6 = 111 - x$$

Step4: Rearrange to solve for x

Rearrange the equation to isolate $x$:
$$x = 111 - 63.6$$

Answer:

$47.4$