Question

let \\( \overline { f g } \\) be an angle bisector. which statement is true?

angle dgf is a right angle.
angle dfg is congruent to angle efg.
dg = ge

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two congruent (equal - measure) angles.
Since \(\overline{FG}\) is an angle bisector of \(\angle DFE\), by the definition of an angle bisector, \(\angle DFG\) and \(\angle EFG\) are the two angles formed by the bisector \(\overline{FG}\) of \(\angle DFE\).
So, \(\angle DFG\cong\angle EFG\)

Step2: Analyze other options

• For the option “Angle \(DGF\) is a right angle”: There is no information given in the problem (such as perpendicular lines or right - angle markings) to suggest that \(\angle DGF = 90^{\circ}\).
• For the option “\(DF = EF\)”: Just because \(\overline{FG}\) is an angle bisector does not imply that the sides \(DF\) and \(EF\) are equal. An angle bisector only relates to the angles it divides, not the adjacent sides (unless in an isosceles triangle, but there is no indication of that here).
• For the option “\(DG=GE\)”: An angle bisector does not necessarily divide the opposite side into two equal segments. The segment - dividing property related to angle bisectors is more complex (the angle - bisector theorem which states \(\frac{DG}{GE}=\frac{DF}{EF}\)) and there is no basis for \(DG = GE\) from the given information that \(\overline{FG}\) is an angle bisector.

Answer:

Angle \(DFG\) is congruent to Angle \(EFG\)