Question

in circle e with $m\angle dgf = 27^\circ$, find the measure of minor arc $\overset{\frown}{df}$.

Explanation:

Step1: Recall inscribed angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. For $\angle DGF$, it intercepts arc $\widehat{DF}$, so $m\angle DGF = \frac{1}{2}m\widehat{DF}$.

Step2: Solve for arc measure

Rearrange the formula to solve for $m\widehat{DF}$: $m\widehat{DF} = 2 \times m\angle DGF$.
Substitute $m\angle DGF = 27^\circ$:
$m\widehat{DF} = 2 \times 27^\circ$

Answer:

$54^\circ$