Question

given: \\( \overline { hf } \parallel \overline { jk } ; \overline { hg } \cong \overline { jg } \\) prove: \\( \triangle f h g \cong \triangle k j g \\) to prove that the triangles are congruent by asa, which statement and reason could be used as part of the proof? \\( \angle f g h \cong \angle k g j \\) because vertical angles are congruent. \\( \angle j k g \cong \angle h f g \\) because vertical angles are congruent. \\( \angle f h g \cong \angle j k g \\) because right angles are congruent. \\( \angle h f g \cong \angle k j g \\) because alternate interior angles are congruent.

Explanation:

• For the ASA (Angle - Side - Angle) congruence criterion, we need two angles and the included side to be congruent.
• Given \( \overline{HF}\parallel\overline{JK}\), by the alternate - interior angles theorem, when two parallel lines are cut by a transversal, the alternate interior angles are congruent. Here, the transversal is \( \overline{GK}\) (or \( \overline{GF}\)) for the parallel lines \( \overline{HF}\) and \( \overline{JK}\).
• We know that \( \angle HFG\) and \( \angle KJG\) are alternate interior angles. Also, we are given \( \overline{HG}\cong\overline{JG}\) (the side). And \( \angle HGF\) and \( \angle JGK\) are vertical angles (so \( \angle HGF\cong\angle JGK\)).

Answer:

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