Question

given ∠abc and ∠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 options: asa congruence theorem, aas congruence theorem, reflexive property, third angle theorem

Explanation:

Step1: Identify Given Information

We know $\angle ABC$ and $\angle FGH$ are right angles (so $\angle ABC \cong \angle FGH$), $\overline{BA} \parallel \overline{GF}$ (so $\angle BAC \cong \angle GFH$ by corresponding angles), and $\overline{BC} \cong \overline{GH}$ (given).

Step2: Determine Congruence Theorem

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

Answer:

AAS congruence theorem.