Question

the proof that $\triangle efg \cong \triangle jhg$ is shown. given: $g$ is the midpoint of $\overline{hf}$, $\overline{ef} \parallel \overline{hj}$, and $\overline{ef} \cong \overline{hj}$. prove: $\triangle efg \cong \triangle jhg$. what is the missing statement in the proof? $\bigcirc \angle feg \cong \angle hjg$; $\bigcirc \angle gfe \cong \angle ghj$; $\bigcirc \angle egf \cong \angle jgh$; $\bigcirc \angle gef \cong \angle jhg$.

Explanation:

To prove \(\triangle EFG \cong \triangle JHG\), we use the given information: \(G\) is the midpoint of \(\overline{HF}\) (so \(\overline{FG} \cong \overline{HG}\)), \(\overline{EF} \parallel \overline{HJ}\) (so alternate interior angles are congruent), and \(\overline{EF} \cong \overline{HJ}\). When \(\overline{EF} \parallel \overline{HJ}\), the alternate interior angles formed by transversal \(\overline{HF}\) are \(\angle GFE\) and \(\angle GHJ\). So \(\angle GFE \cong \angle GHJ\) is the missing statement (to use SAS congruence: \(\overline{EF} \cong \overline{HJ}\), \(\angle GFE \cong \angle GHJ\), \(\overline{FG} \cong \overline{HG}\)).

Answer:

\(\boldsymbol{\angle GFE \cong \angle GHJ}\) (corresponding option: the second one, e.g., if options are labeled as A, B, C, D, then B. \(\angle GFE \cong \angle GHJ\))