Question

given: $overline{hf}paralleloverline{jk};overline{hg}congoverline{jg}$ prove: $\triangle fhgcong\triangle kjg$ to prove that the triangles are congruent by asa, which statement and reason could be used as part of the proof? $angle fghcongangle kgj$ because vertical angles are congruent. $angle jkgcongangle hfg$ because vertical angles are congruent. $angle fhgcongangle jkg$ because right angles are congruent. $angle hfgcongangle kjg$ because alternate interior angles are congruent.

Explanation:

Step1: Recall parallel - line angle properties

Since $\overline{HF}\parallel\overline{JK}$, when a transversal (in this case, the line containing $\overline{HG}$ and $\overline{JG}$) intersects two parallel lines, alternate interior angles are congruent.

Step2: Identify alternate - interior angles

In the given figure, $\angle HFG$ and $\angle KJG$ are alternate - interior angles formed by the parallel lines $\overline{HF}$ and $\overline{JK}$ and the transversal.

Step3: Recall ASA congruence criterion

The ASA (Angle - Side - Angle) congruence criterion states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. We already know that $\overline{HG}\cong\overline{JG}$, and we need appropriate angle - congruence statements. The congruence of $\angle HFG$ and $\angle KJG$ (alternate interior angles) can be used as part of the ASA proof.

Answer:

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