Question

find the value of ( x ), where ( x^circ = moverarc{fk} ).

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:
$$m\angle H = \frac{1}{2}(m\overset{\frown}{FK} - m\overset{\frown}{GJ})$$

Step2: Substitute known values

We know $m\angle H = 29.4^\circ$, $m\overset{\frown}{GJ}=69.2^\circ$, and $m\overset{\frown}{FK}=x^\circ$. Plug these into the formula:
$$29.4 = \frac{1}{2}(x - 69.2)$$

Step3: Solve for $x$ (multiply both sides by 2)

$$29.4 \times 2 = x - 69.2$$
$$58.8 = x - 69.2$$

Step4: Isolate $x$ (add 69.2 to both sides)

$$x = 58.8 + 69.2$$
$$x = 128$$

Answer:

$128$