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}
\triangle efg and \triangle hig are right triangles
\triangle efg cong \triangle hig
reasons

Explanation:

Step1: Identify Midpoint Property

G is the midpoint of \( \overline{EH} \), so by definition of midpoint, \( EG \cong HG \).

Step2: Identify Right Angles

\( \angle E \) and \( \angle H \) are right angles, so \( \angle E \cong \angle H \) (all right angles are congruent).

Step3: Identify Given Congruent Sides

Given \( FG \cong GI \) (wait, likely a typo, should be \( FG \cong HG \)? No, the diagram shows vertical angles? Wait, actually, \( \angle EGF \) and \( \angle HGI \) are vertical angles, so \( \angle EGF \cong \angle HGI \) (vertical angles congruent). Wait, the given is \( FG \cong GI \)? No, maybe \( FG \cong HG \)? Wait, the problem says \( FG \cong GI \), but maybe it's \( FG \cong HG \)? Wait, no, let's re - examine. The triangles are \( \triangle EFG \) and \( \triangle HIG \)? Wait, the right angles are \( \angle E \) and \( \angle H \), so \( \triangle EFG \) and \( \triangle HIG \) are right triangles. We have \( EG \cong HG \) (midpoint), \( \angle E \cong \angle H \) (right angles), and \( \angle EGF \cong \angle HGI \) (vertical angles). Wait, but the given is \( FG \cong GI \). Wait, maybe the triangles are \( \triangle EFG \) and \( \triangle HIG \), with \( FG \cong IG \), \( EG \cong HG \), and right angles at \( E \) and \( H \). So by SAS (Side - Angle - Side) congruence criterion, \( \triangle EFG \cong \triangle HIG \).

Step4: List Statements and Reasons

Statements	Reasons
2. \( EG \cong HG \)	Definition of midpoint
3. \( \angle E \) and \( \angle H \) are right angles	Given
4. \( \angle E \cong \angle H \)	All right angles are congruent
5. \( FG \cong GI \)	Given
6. \( \angle EGF \cong \angle HGI \)	Vertical angles are congruent
7. \( \triangle EFG \cong \triangle HIG \)	SAS ( \( EG \cong HG \), \( \angle E \cong \angle H \), \( FG \cong GI \))

Answer:

The proof is completed by showing \( \triangle EFG \cong \triangle HIG \) using the SAS congruence criterion with \( EG \cong HG \) (midpoint), \( \angle E \cong \angle H \) (right angles), and \( FG \cong GI \) (given), along with vertical angles \( \angle EGF \cong \angle HGI \).