Question

angle g is a circumscribed angle of circle e. major arc fd measures $280^\circ$. what is the measure of angle gfd?
$40^\circ$
$50^\circ$
$80^\circ$
$90^\circ$

Explanation:

Step1: Find minor arc FD

The total degrees in a circle are $360^\circ$. Subtract the major arc measure from $360^\circ$.
$\text{Minor arc } FD = 360^\circ - 280^\circ = 80^\circ$

Step2: Find $\angle FED$

The central angle equals its intercepted arc.
$\angle FED = \text{Minor arc } FD = 80^\circ$

Step3: Find $\angle G$

For a circumscribed angle, $\angle G = \frac{1}{2}(\text{Major arc } FD - \text{Minor arc } FD)$
$\angle G = \frac{1}{2}(280^\circ - 80^\circ) = 100^\circ$

Step4: Recognize GF and GD are tangents

Tangents from a point to a circle are equal, so $GF = GD$, making $\triangle GFD$ isosceles. Thus $\angle GFD = \angle GDF$.

Step5: Calculate $\angle GFD$

Sum of angles in a triangle is $180^\circ$. Solve for the base angle.
$\angle GFD = \frac{180^\circ - \angle G}{2} = \frac{180^\circ - 100^\circ}{2} = 40^\circ$

Answer:

40°