Question

given: ∠abc and ∠fgh are right angles; $overline{ba} parallel overline{cf}$; $overline{bc} cong overline{gh}$ prove: $\triangle abc cong \triangle fgh$ step 1: we know that ∠abc ≅ ∠fgh because all right angles are congruent. step 2: we know that ∠bac ≅ ∠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:

1. Recall the congruence theorems:

• ASA (Angle - Side - Angle) requires two angles and the included side to be congruent.
• AAS (Angle - Angle - Side) requires two angles and a non - included side to be congruent.
• Third angle theorem is about the third angle being congruent if two angles of one triangle are congruent to two angles of another triangle, and it is not a congruence theorem for triangles.
• Reflexive property states that a segment or angle is congruent to itself, which is not relevant here.

1. Analyze the given information:

• We have \(\angle ABC\cong\angle FGH\) (right angles), \(\angle BAC\cong\angle GFH\) (corresponding angles of parallel lines), and \(\overline{BC}\cong\overline{GH}\) (given).
• In \(\triangle ABC\) and \(\triangle FGH\), we have two angles (\(\angle ABC\cong\angle FGH\) and \(\angle BAC\cong\angle GFH\)) and a non - included side (\(\overline{BC}\cong\overline{GH}\)) congruent. This matches the AAS (Angle - Angle - Side) congruence theorem.

Answer:

AAS congruence theorem.