Question

given: \\( \angle abc \\) and \\( \angle fgh \\) are right angles; \\( \overline{ba} \parallel \overline{gf} \\); \\( \overline{bc} \cong \overline{gh} \\) prove: \\( \triangle abc \cong \triangle fgh \\) step 1: we know that \\( \angle abc \cong \angle fgh \\) because all right angles are congruent. step 2: we know that \\( \angle bac \cong \angle gfh \\) because corresponding angles of parallel lines are congruent. step 3: we know that \\( \overline{bc} \cong \overline{gh} \\) because it is given. step 4: \\( \triangle abc \cong \triangle fgh \\) because of the asa congruence theorem. aas congruence theorem. third angle theorem. reflexive property.

Explanation:

Step 1: Identify the given information

We know that \(\angle ABC\cong\angle FCH\) (all right angles are congruent), \(\overline{BC}\cong\overline{GH}\) (given), and \(\angle BAC\cong\angle GFH\) (corresponding angles of parallel lines \(BA\parallel GF\) are congruent).

Step 2: Recall triangle congruence theorems

The AAS (Angle - Angle - Side) congruence theorem states that if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
In \(\triangle ABC\) and \(\triangle FGH\), we have two angles (\(\angle ABC\cong\angle FCH\) and \(\angle BAC\cong\angle GFH\)) and a non - included side (\(\overline{BC}\cong\overline{GH}\)).

Answer:

AAS congruence theorem.