Question

$\overrightarrow{fe}$ is tangent to the circle at $e$. $\overrightarrow{fg}$ is tangent to the circle at $g$. find the measure of $\angle efg$. write your answer as a whole number or a decimal. $m\angle efg = \square^{\circ}$

Explanation:

Step1: Recall tangent-chord angle theorem

The measure of an angle formed by two tangents outside a circle is half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc \(EG\) (98°) and the minor arc \(EG\). First, calculate the minor arc \(EG\):
$$360^\circ - 98^\circ = 262^\circ$$

Step2: Calculate angle ∠EFG

Use the formula for the angle outside the circle:
$$m\angle EFG = \frac{1}{2} \times (\text{measure of major arc } EG - \text{measure of minor arc } EG)$$
Substitute the values:
$$m\angle EFG = \frac{1}{2} \times (262^\circ - 98^\circ)$$

Step3: Simplify the expression

First compute the difference inside the parentheses:
$$262^\circ - 98^\circ = 164^\circ$$
Then multiply by \(\frac{1}{2}\):
$$\frac{1}{2} \times 164^\circ = 82^\circ$$

Answer:

$m\angle EFG = 82^\circ$