Question

given g is the midpoint of eh,
fg ≅ gi, ∠e and ∠h are right angles.
prove △efg ≅ △hig
g is the midpoint of eh, fg ≅ gi, ∠e and ∠h are right angles.
eg ≅ hg
△efg and △hig are right triangles.
△efg ≅ △hig

Explanation:

Step1: Definition of midpoint

Since \(G\) is the midpoint of \(EH\), by the definition of midpoint, \(EG = GH\) (or \(\overline{EG}\cong\overline{HG}\)).

Step2: Right - angle definition

\(\angle E\) and \(\angle H\) are right angles. So, \(\angle E=\angle H = 90^{\circ}\).

Step3: Vertical angles

\(\angle EGF\) and \(\angle HGI\) are vertical angles. By the vertical - angles theorem, \(\angle EGF\cong\angle HGI\).

Step4: \(SAS\) (Side - Angle - Side) congruence criterion

In \(\triangle EFG\) and \(\triangle HIG\), we have \(\overline{EG}\cong\overline{HG}\) (from Step 1), \(\angle EGF\cong\angle HGI\) (from Step 3), and \(\overline{FG}\cong\overline{GI}\) (given). So, by the \(SAS\) congruence criterion, \(\triangle EFG\cong\triangle HIG\).

Answer:

STATEMENTS	REASONS
\(\overline{EG}\cong\overline{HG}\)	Definition of midpoint
\(\triangle EFG\) and \(\triangle HIG\) are right triangles	Definition of right triangle (\(\angle E = \angle H=90^{\circ}\))
\(\angle EGF\cong\angle HGI\)	Vertical - angles theorem
\(\triangle EFG\cong\triangle HIG\)	\(SAS\) (Side - Angle - Side) congruence criterion (\(\overline{EG}\cong\overline{HG}\), \(\angle EGF\cong\angle HGI\), \(\overline{FG}\cong\overline{GI}\))