Question

find the value of ( x ). note that, in the image below, ( x^circ = moverarc{ad} ), ( m angle c = 34^circ ), and ( moverarc{bd} = 59^circ ).

Explanation:

Step1: Recall secant-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:
$$m\angle C = \frac{1}{2}(m\overset{\frown}{AD} - m\overset{\frown}{BD})$$

Step2: Substitute known values

Plug $m\angle C=34^\circ$, $m\overset{\frown}{BD}=59^\circ$, $m\overset{\frown}{AD}=x^\circ$ into the formula:
$$34 = \frac{1}{2}(x - 59)$$

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

$$34 \times 2 = x - 59$$
$$68 = x - 59$$

Step4: Isolate $x$ (add 59 to both sides)

$$x = 68 + 59$$
$$x = 127$$

Answer:

$127$