Question

triangle congruence
given: hf || jk; hg ≅ jg
prove: △fhg ≅ △kjg
to prove that the triangles are congruent by asa, which statement and reason could be used as part of the proof?
∠hfg ≅ ∠kjg because alternate interior angles are congruent.
∠fgh ≅ ∠kgj because vertical angles are congruent.
∠fhg ≅ ∠jkg because right angles are congruent.
∠jkg ≅ ∠hfg because vertical angles are congruent.

Explanation:

Step1: Recall ASA (Angle - Side - Angle) criterion

ASA requires two pairs of congruent angles and the included side between them to be congruent. We know that $\overline{HG}\cong\overline{JG}$ (given). We need two appropriate angle - congruence statements.

Step2: Analyze angle relationships

Since $\overline{HF}\parallel\overline{JK}$, by the property of alternate - interior angles, $\angle HFG\cong\angle KJG$. Also, $\angle FGH$ and $\angle KGJ$ are vertical angles. Vertical angles are always congruent. For ASA, we need the angles that are adjacent to the given congruent side $\overline{HG}\cong\overline{JG}$. The vertical angles $\angle FGH\cong\angle KGJ$ are the ones that can be used along with the given side - congruence and the alternate - interior angle congruence for the ASA proof.

Answer:

$\angle FGH\cong\angle KGJ$ because vertical angles are congruent.