Question

in the diagram, which is not drawn to scale, g is the incenter of △def. m∠efd = 48°, and m∠edf = 2·m∠def. find m∠deg.
m∠deg = \boxed{\circ}
(type an integer or a decimal.)

Explanation:

Step1: Find \(m\angle EDF\)

Since \(G\) is the in - center of \(\triangle DEF\), \(DG\), \(EG\), \(FG\) are angle bisectors.
We know that \(m\angle EFD = 48^{\circ}\), and by the angle - sum property of a triangle \(\angle DEF+\angle EDF+\angle EFD=180^{\circ}\). Let \(m\angle DEF = x\), then \(m\angle EDF = 2x\).
So \(x + 2x+48^{\circ}=180^{\circ}\).
Combining like terms: \(3x=180^{\circ}- 48^{\circ}=132^{\circ}\).
Solving for \(x\): \(x=\frac{132^{\circ}}{3}=44^{\circ}\). So \(m\angle DEF = 44^{\circ}\).

Step2: Find \(m\angle DEG\)

Since \(EG\) is the angle bisector of \(\angle DEF\), \(m\angle DEG=\frac{1}{2}m\angle DEF\).
Substituting \(m\angle DEF = 44^{\circ}\), we get \(m\angle DEG=\frac{1}{2}\times44^{\circ}\).

Answer:

\(22\)