Question

h is the midpoint of \\(\overline{fg}\\) and \\(\overline{fg} \perp \overline{eh}\\). complete the proof that \\(\overline{eg} \cong \overline{ef}\\).

(image of a triangle efg with h on fg, eh connecting to h)

statement	reason
2 \\(\overline{fg} \perp \overline{eh}\\)	given
3 \\(\angle ehf \cong \angle ehg\\)	all right angles are congruent
4 \\(\overline{fh} \cong \overline{gh}\\)	definition of midpoint
5 \\(\overline{eh} \cong \overline{eh}\\)	reflexive property of congruence
6

Explanation:

Step1: Prove triangles congruent

Since \( \angle EHF\cong\angle EHG\), \( \overline{FH}\cong\overline{GH}\), and \( \overline{EH}\cong\overline{EH}\), by the Side - Angle - Side (SAS) congruence criterion, \( \triangle EHF\cong\triangle EHG\).

Step2: Use congruent triangles property

If two triangles \( \triangle EHF\) and \( \triangle EHG\) are congruent, then their corresponding sides are congruent. So, \( \overline{EG}\cong\overline{EF}\) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Answer:

\( \triangle EHF\cong\triangle EHG\) (by SAS); \( \overline{EG}\cong\overline{EF}\) (by CPCTC)