Question

partially correct
your answer is incorrect.

• m∠dxg: your answer is incorrect.

the angle bisectors of δxyz are xg, yg, and zg. they meet at a single point g.
(in other words, g is the incenter of δxyz.)
suppose eg = 8, yg = 13, m∠ezf = 34°, and m∠dyg = 58°.
find the following measures.
note that the figure is not drawn to scale.
fg = 8
m∠dye = 116°
m∠dxg = 90°
try one last time

Explanation:

Step1: Analyze angle bisector property

$G$ is the incenter, so $ZG$ bisects $\angle EZF$. Thus, $m\angle EZG = m\angle FZG = \frac{34^\circ}{2} = 17^\circ$. $YG$ bisects $\angle DYE$, so $m\angle DYG = m\angle EYG = 58^\circ$, hence $m\angle DYE = 58^\circ \times 2 = 116^\circ$ (correct as given). $EG, FG, DG$ are inradii, so $EG = FG = DG = 8$ (since $EG=8$, $FG=8$ is correct).

Step2: Find $m\angle DXG$

First, find $\angle XYZ = 116^\circ$ (from $\angle DYE$), $\angle XZY = 34^\circ$. Sum of angles in $\triangle XYZ$: $\angle YXZ = 180^\circ - 116^\circ - 34^\circ = 30^\circ$. Since $XG$ bisects $\angle YXZ$, $m\angle DXG = \frac{30^\circ}{2} = 15^\circ$.

Answer:

$FG = 8$
$m\angle DYE = 116^\circ$
$m\angle DXG = 15^\circ$