Question

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? ∠fgh ≅ ∠kgj because vertical angles are congruent. ∠jkg ≅ ∠hfg because vertical angles are congruent. ∠fhg ≅ ∠jkg because right angles are congruent. ∠hfg ≅ ∠kjg because alternate interior angles are congruent.

Explanation:

Step1: Recall ASA congruence criterion

ASA (Angle - Side - Angle) requires two pairs of congruent angles and the included side congruent. We know $\overline{HG}\cong\overline{JG}$ (given side). We need appropriate angle - angle pairs.

Step2: Analyze angle relationships

Since $\overline{HF}\parallel\overline{JK}$, by the property of alternate - interior angles, when two parallel lines are cut by a transversal, alternate - interior angles are congruent. $\angle HFG$ and $\angle KJG$ are alternate - interior angles formed by the parallel lines $\overline{HF}$ and $\overline{JK}$ and the transversal $\overline{FJ}$.

Step3: Evaluate other angle options

• $\angle FGH\cong\angle KGJ$ is about vertical angles, but this doesn't help with ASA as it's not the correct pair for the given parallel lines and side.
• $\angle JKG$ and $\angle HFG$ are not vertical angles.
• There is no information to suggest $\angle FHG$ and $\angle JKG$ are right angles.

Answer:

$\angle HFG\cong\angle KJG$ because alternate interior angles are congruent.