Question

$\angle g \cong \angle h$. complete the proof that $\triangle fgh \cong \triangle fhg$.
(image of triangle with vertices g, h, f)

	statement	reason
2	$\overline{gh} \cong \overline{gh}$
3	$\triangle fgh \cong \triangle fhg$

Explanation:

Step1: Identify Given Information

The problem states that \( \angle G \cong \angle H \), so the reason for this statement is "Given".

Step2: Identify Common Side

For \( \overline{GH} \cong \overline{GH} \), this is the Reflexive Property of Congruence (a segment is congruent to itself).

Step3: Identify Congruence Criterion

To prove \( \triangle FGH \cong \triangle FHG \), we have \( \angle G \cong \angle H \) (given), \( \overline{GH} \cong \overline{GH} \) (reflexive), and we can use the AAS (Angle - Angle - Side) or ASA (Angle - Side - Angle) congruence criterion? Wait, actually, let's look at the triangles. In \( \triangle FGH \) and \( \triangle FHG \), we have \( \angle G \cong \angle H \), \( \overline{GH} \cong \overline{GH} \), and also \( \angle F \) is common? Wait, no, let's re - examine. Wait, the triangles are \( \triangle FGH \) and \( \triangle FHG \). The sides and angles: \( \angle G \cong \angle H \), side \( GH \) is common, and also \( \angle F \) is the same angle for both triangles? Wait, no, actually, when we have two angles and a side. Wait, the correct congruence criterion here is AAS? Wait, no, let's see: in \( \triangle FGH \) and \( \triangle FHG \), we have \( \angle G \cong \angle H \), \( \overline{GH} \cong \overline{GH} \), and \( \angle F \cong \angle F \) (reflexive). Wait, no, maybe it's ASA? Wait, no, the correct criterion here, since we have two angles and the included side? Wait, no, \( \angle G \), side \( GH \), and \( \angle F \) in \( \triangle FGH \); and \( \angle H \), side \( GH \), and \( \angle F \) in \( \triangle FHG \). So actually, the congruence criterion is AAS (Angle - Angle - Side) or ASA? Wait, no, the correct reason for \( \triangle FGH \cong \triangle FHG \) is AAS? Wait, no, let's check the order. Wait, the given angle is \( \angle G \cong \angle H \), the common side is \( GH \), and the other angle: \( \angle F \) is common. So actually, it's ASA? Wait, no, maybe I made a mistake. Wait, the triangles are named \( \triangle FGH \) and \( \triangle FHG \). So vertex \( F \) corresponds to \( F \), \( G \) corresponds to \( H \), and \( H \) corresponds to \( G \). So \( \angle G \cong \angle H \) (given), \( \overline{GH} \cong \overline{GH} \) (reflexive), and \( \angle F \cong \angle F \) (reflexive). Wait, no, that would be AAA, which doesn't prove congruence. Wait, I think I messed up. Wait, no, let's look at the sides. Wait, in \( \triangle FGH \), the sides are \( FG \), \( GH \), \( FH \); in \( \triangle FHG \), the sides are \( FH \), \( HG \), \( FG \). Wait, maybe it's SAS? No, we have an angle. Wait, the correct congruence criterion here is AAS. Wait, no, the correct reason is that we have two angles and a non - included side? Wait, no, let's start over.

1. Statement 1: \( \angle G \cong \angle H \), Reason: Given
2. Statement 2: \( \overline{GH} \cong \overline{GH} \), Reason: Reflexive Property of Congruence (a segment is congruent to itself)
3. Statement 3: \( \triangle FGH \cong \triangle FHG \), Reason: AAS (Angle - Angle - Side) or ASA? Wait, no, actually, when we have \( \angle G \cong \angle H \), \( \overline{GH} \cong \overline{GH} \), and \( \angle F \cong \angle F \), but that's AAA. Wait, I think I made a mistake in the triangle naming. Wait, the triangles are \( \triangle FGH \) and \( \triangle FHG \). Let's list the corresponding parts:

- In \( \triangle FGH \): angles \( \angle F \), \( \angle G \), \( \angle H \); sides \( FG \), \( GH \), \( FH \)
- In \( \triangle FHG \): angles \( \angle F \), \( \angle H \), \( \angle G \); sides \( FH \), \( HG \), \( FG \)

So we have \( \angle G \cong \angle H \) (given), \( \overline{GH} \cong \overline{GH} \) (reflexive), and \( \angle F \cong \angle F \) (reflexive). Wait, no, that's not right. Wait, maybe the correct congruence criterion is AAS. Wait, no, the correct reason for the congruence of the triangles is AAS (Angle - Angle - Side) or ASA? Wait, I think the correct congruence criterion here is AAS. Wait, no, let's check the order. The two angles are \( \angle G \) and \( \angle F \), and the side \( GH \) in \( \triangle FGH \); and \( \angle H \) and \( \angle F \), and the side \( GH \) in \( \triangle FHG \). So it's AAS (Angle - Angle - Side) because we have two angles and a non - included side.

So to summarize:

1. Reason for \( \angle G \cong \angle H \): Given
2. Reason for \( \overline{GH} \cong \overline{GH} \): Reflexive Property of Congruence
3. Reason for \( \triangle FGH \cong \triangle FHG \): AAS (Angle - Angle - Side) Congruence Criterion (or ASA? Wait, no, AAS is more appropriate here as we have two angles and a non - included side)

Answer:

1. Reason: Given
2. Reason: Reflexive Property of Congruence
3. Reason: AAS (Angle - Angle - Side) Congruence Criterion