Question

h is the midpoint of (overline{fg}). complete the proof that (\triangle eghcong\triangle efh).

Explanation:

Step1: Given

Statement 1 is given in the problem.

Step2: Given

Statement 2 is given in the problem (from the right - angle symbol in the diagram).

Step3: Definition of perpendicular lines

Since \(EH\perp FG\), \(\angle EHF = 90^{\circ}\) and \(\angle EHG=90^{\circ}\). So, \(\angle EHF\cong\angle EHG\) (all right angles are congruent).

Step4: Definition of midpoint

If \(H\) is the midpoint of \(FG\), then \(FH = GH\) (by the definition of a midpoint, which divides a segment into two equal parts). So, \(\overline{FH}\cong\overline{GH}\).

Step5: Reflexive property of congruence

For any segment \(AB\), \(AB\cong AB\). So, \(\overline{EH}\cong\overline{EH}\).

Step6: SAS (Side - Angle - Side) congruence criterion

In \(\triangle EGH\) and \(\triangle EFH\), we have \(\overline{FH}\cong\overline{GH}\) (from step 4), \(\angle EHF\cong\angle EHG\) (from step 3), and \(\overline{EH}\cong\overline{EH}\) (from step 5). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle EGH\cong\triangle EFH\).

Answer:

1. Given
2. Given
3. Definition of perpendicular lines (all right angles are congruent)
4. Definition of midpoint
5. Reflexive property of congruence
6. SAS (Side - Angle - Side) congruence criterion