Question

∠j ≅ ∠h and \overline{hk} ≅ \overline{gj}. complete the proof that \triangle fgj ≅ \triangle fkh.

Explanation:

To prove $\Delta FGJ \cong \Delta FKH$, we use the ASA congruence criterion. First, we identify the vertical angles formed by intersecting lines $\overline{JH}$ and $\overline{FK}$: $\angle JIG \cong \angle HIK$, but more directly, the included angles at $F$ are the same (common angle). Wait, correction: The triangles share $\angle F$ (common angle). So we have $\angle J \cong \angle H$ (given), $\overline{JG} \cong \overline{HK}$ (given, note: $\overline{HK} \cong \overline{GJ}$ is same as $\overline{JG} \cong \overline{HK}$), and $\angle JFG \cong \angle HFK$ (common angle). Thus ASA applies.

Answer:

$\Delta FGJ \cong \Delta FKH$ by ASA congruence (common angle $\angle F$, $\angle J \cong \angle H$, $\overline{JG} \cong \overline{HK}$).