Question

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

1 ∠j ≅ ∠h given
2 \overline{hk} ≅ \overline{gj} given
3

Explanation:

Step1: Identify vertical angles

$\angle JIG \cong \angle HIK$ (Vertical angles are congruent)

Step2: Prove triangle congruence

$\triangle JIG \cong \triangle HIK$ (AAS: $\angle J \cong \angle H$, $\angle JIG \cong \angle HIK$, $\overline{GJ} \cong \overline{HK}$)

Step3: Deduce corresponding sides

$\overline{JI} \cong \overline{HI}$ and $\overline{GI} \cong \overline{KI}$ (CPCTC)

Step4: Find shared angle

$\angle JFG = \angle HFK$ (Common angle)

Step5: Prove final congruence

$\triangle FGJ \cong \triangle FKH$ (ASA: $\angle J \cong \angle H$, $\overline{GJ} \cong \overline{HK}$, $\angle JFG \cong \angle HFK$)

Answer:

$\triangle FGJ \cong \triangle FKH$ by ASA congruence criterion.