Question

in the diagram, which is not drawn to scale, g is the incenter of $\triangle def$, $m\angle def = 54^{\circ}$, and $m\angle efd = 2\cdot m\angle edf$. find $m\angle fdg$.
$m\angle fdg = \square^{\circ}$
(type an integer or a decimal.)

Explanation:

Step1: Set variable for $\angle EDF$

Let $m\angle EDF = x$. Then $m\angle EFD = 2x$.

Step2: Use triangle angle sum theorem

Sum of angles in $\triangle DEF$ is $180^\circ$.
$$x + 2x + 54^\circ = 180^\circ$$

Step3: Solve for $x$

Simplify and isolate $x$:
$$3x = 180^\circ - 54^\circ = 126^\circ$$
$$x = \frac{126^\circ}{3} = 42^\circ$$

Step4: Use incenter property

The incenter bisects $\angle EDF$, so $m\angle FDG = \frac{1}{2}m\angle EDF$.
$$m\angle FDG = \frac{1}{2} \times 42^\circ = 21^\circ$$

Answer:

$21$