Question

1. given: \\( \overline { e g } \\) is the perpendicular bisector of \\( \overline { f h } \\) prove \\( \triangle f g e \cong \triangle h g e \\) statements justifications 1. 1. given 2. \\( \overline { f g } \cong \overline { h g } \\) 2. definition of perpendicular bisector 3. \\( \overline { e g } \cong \overline { e g } \\) 3. 4. 4. definition of perpendicular bisector 5. \\( \triangle f g e \cong \triangle h g e \\) 5.

Explanation:

Step1: Identify the given information

We are given that \( \overline{EG} \) is the perpendicular bisector of \( \overline{FH} \).

Step2: Analyze statement 3

For statement 3, \( \overline{EG}\cong\overline{EG} \) by the reflexive property of congruence (a segment is congruent to itself).

Step3: Analyze statement 4

Since \( \overline{EG} \) is the perpendicular bisector of \( \overline{FH} \), \( \angle EGF=\angle EGH = 90^{\circ}\) (by the definition of perpendicular bisector).

Step4: Prove the triangles congruent

We have \( \overline{FG}\cong\overline{HG}\) (from statement 2, definition of perpendicular bisector), \( \overline{EG}\cong\overline{EG}\) (reflexive property), and \( \angle EGF=\angle EGH\) (from statement 4, definition of perpendicular bisector). So, by the Side - Angle - Side (SAS) congruence criterion, \( \triangle FGE\cong\triangle HGE\).

Answer:

1. \( \overline{EG} \) is the perpendicular bisector of \( \overline{FH} \)
2. \( \overline{FG}\cong\overline{HG} \)
3. Reflexive property of congruence
4. \( \angle EGF=\angle EGH = 90^{\circ}\)
5. SAS (Side - Angle - Side) congruence criterion