In the figure below, triangle DEF is an equilateral triangle Point G is the incenter of the triangle. Find measure angle DGF.
In an equilateral triangle, all angles are congruent and measure 60 degrees. Since triangle DEF is an equilateral triangle, angle DEF measures 60 degrees.
The incenter of a triangle is the point where the angle bisectors intersect. In an equilateral triangle, the angle bisectors also serve as medians and altitudes. This means that the incenter is also the centroid and the orthocenter.
Since G is the incenter of triangle DEF, the angle DGF is formed by the angle bisector of angle DEF. Since an angle bisector cuts an angle into two congruent angles, angle DGF must measure half of angle DEF.
Therefore, angle DGF measures:
(1/2) * 60 degrees = 30 degrees