Question

1 h is the midpoint of \\( \overline { f g } \\) 2 \\( \overline { e h } \perp \overline { f g } \\) 3 \\( \angle e h f \cong \angle e h g \\) 4 \\( \overline { f h } \cong \overline { g h } \\) 5 \\( \overline { e h } \cong \overline { e h } \\) 6 \\( \delta e f h \cong \delta e g h \\)

Explanation:

Step1: Identify the given information

Statement 1: \(H\) is the midpoint of \(\overline{FG}\) (Given).
Statement 2: \(\overline{EH}\perp\overline{FG}\) (Given).

Step2: Analyze the right - angle property

Since \(\overline{EH}\perp\overline{FG}\), \(\angle EHF\) and \(\angle EHG\) are right angles. By the definition of right angles, \(\angle EHF\cong\angle EHG\) (All right angles are congruent).

Step3: Use the mid - point property

If \(H\) is the midpoint of \(\overline{FG}\), then by the definition of a midpoint, \(FH = GH\), so \(\overline{FH}\cong\overline{GH}\).

Step4: Use the reflexive property

For any segment \(\overline{AB}\), \(\overline{AB}\cong\overline{AB}\). So for segment \(\overline{EH}\), \(\overline{EH}\cong\overline{EH}\) (Reflexive property of congruence).

Step5: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle EFH\) and \(\triangle EGH\), we have \(\overline{FH}\cong\overline{GH}\) (from Step 3), \(\angle EHF\cong\angle EHG\) (from Step 2), and \(\overline{EH}\cong\overline{EH}\) (from Step 4). By the SAS congruence criterion, \(\triangle EFH\cong\triangle EGH\).

Answer:

1. Given
2. Given
3. All right angles are congruent
4. Definition of mid - point
5. Reflexive property of congruence
6. SAS (Side - Angle - Side) congruence criterion