Question

let \\( \overline { f g } \\) be a median. which statement is true?
angle dfg is congruent to angle efg
\\( d f = e f \\)
angle dgf is a right angle.
\\( d g = g e \\)

Explanation:

Step1: Recall the definition of a median

A median of a triangle is a line segment that joins a vertex to the mid - point of the opposite side. If \(\overline{FG}\) is a median, then \(G\) is the mid - point of \(\overline{DE}\).

Step2: Analyze each option

• Option 1: Angle \(DFG\) is congruent to Angle \(EFG\)

A median does not necessarily bisect the angle at the vertex. So, we cannot say \(\angle DFG\cong\angle EFG\) just because \(FG\) is a median.

• Option 2: \(DF = EF\)

A median connects a vertex to the mid - point of the opposite side. There is no information to suggest that \(DF = EF\). \(DF\) and \(EF\) are two sides of the triangle, and a median does not imply equality of these two sides.

• Option 3: Angle \(DGF\) is a right angle

A median does not imply that it is perpendicular to the side. There is no information given about the angle \(\angle DGF\) being a right angle. A median is just a line segment from a vertex to the mid - point of the opposite side.

• Option 4: \(DG=GE\)

Since \(G\) is the mid - point of \(DE\) (by the definition of a median, if \(FG\) is a median of \(\triangle DEF\) with \(FG\) connecting vertex \(F\) to side \(DE\)), by the definition of a mid - point, \(DG = GE\).

Answer:

\(DG = GE\)