Question

3 segment ge is an angle bisector of both angle hef and angle fgh. prove triangle hge is congruent to triangle fge.

Explanation:

Step1: Identify Angle Bisector Properties

Since \( GE \) bisects \( \angle HEF \), we have \( \angle HEG=\angle FEG \). Since \( GE \) bisects \( \angle FGH \), we have \( \angle HGE = \angle FGE \).

Step2: Identify Common Side

The segment \( GE \) is common to both \( \triangle HGE \) and \( \triangle FGE \), so \( GE = GE \) (reflexive property).

Step3: Apply ASA Congruence Criterion

In \( \triangle HGE \) and \( \triangle FGE \):

• \( \angle HEG=\angle FEG \) (from angle bisector of \( \angle HEF \))
• \( GE = GE \) (common side)
• \( \angle HGE=\angle FGE \) (from angle bisector of \( \angle FGH \))

By the Angle - Side - Angle (ASA) congruence postulate, \( \triangle HGE\cong\triangle FGE \).

Answer:

By the ASA congruence postulate (since \( \angle HEG=\angle FEG \), \( GE = GE \), and \( \angle HGE=\angle FGE \)), \( \triangle HGE\cong\triangle FGE \).