Question

complete the proof
given g is the midpoint of \overline{eh}, \overline{fg} \cong \overline{gi}, \angle e and \angle h are right angles.
prove \triangle efg \cong \triangle hig
statements
g is the midpoint of \overline{eh}, \overline{fg} \cong \overline{gi}, \angle e and \angle h are right angles
\overline{eg} \cong \overline{hg}
\angle efg and \angle hig are right triangles
\angle efg \cong \angle hig
reasons
given
definition of right angles

Explanation:

Step1: Identify Midpoint Property

Since \( G \) is the midpoint of \( EH \), by the definition of a midpoint, \( EG = HG \).

Step2: Identify Right Angles

Given \( \angle E \) and \( \angle H \) are right angles, so \( \angle E \cong \angle H = 90^\circ \).

Step3: Identify Given Side

We know \( FG \cong GI \) (given).

Step4: Apply AAS or HL

We have two right triangles \( \triangle EFG \) and \( \triangle HIG \), with \( EG = HG \) (midpoint), \( \angle E \cong \angle H \) (right angles), and we can also note vertical angles \( \angle EGF \cong \angle HGI \). By AAS (Angle - Angle - Side) congruence criterion, \( \triangle EFG \cong \triangle HIG \).

Answer:

The proof is completed by using the midpoint property, right angle definition, and AAS congruence criterion to show \( \triangle EFG \cong \triangle HIG \). The key steps involve identifying \( EG = HG \) (midpoint), \( \angle E \cong \angle H \) (right angles), and using the given \( FG \cong GI \) (or vertical angles) to apply AAS.