Question

g is the incenter, or point of concurrency, of the angle bisectors of δace. which statements must be true regarding the diagram? □ (overline{bg}congoverline{ag}) □ (overline{dg}congoverline{fg}) □ (overline{dg}congoverline{bg}) □ (overline{ge}) bisects (angle def) □ (overline{ga}) bisects (angle baf)

Explanation:

Step1: Recall in - center property

The in - center of a triangle is equidistant from the sides of the triangle. Since \(G\) is the in - center of \(\triangle ACE\), and \(BG\perp AC\), \(DG\perp CE\), \(FG\perp AE\), the distances from \(G\) to the sides of the triangle are equal. That is, \(DG = FG=BG\).

Step2: Analyze angle - bisector property

There is no information given in the problem to suggest that \(GE\) bisects \(\angle DEF\) or \(GA\) bisects \(\angle BAF\). Also, there is no reason for \(\overline{BG}\cong\overline{AG}\) just because \(G\) is the in - center of \(\triangle ACE\).

Answer:

\(\overline{DG}\cong\overline{FG}\), \(\overline{DG}\cong\overline{BG}\)