Question

$$ \overleftrightarrow { vx } $$ bisects $$ \angle w v y $$ and $$ \angle w \cong \angle y $$. complete the proof that $$ \triangle v x y \cong \triangle v x w $$.

statement	reason
2 $$ \angle w \cong \angle y $$	given
3 $$ \angle w v x \cong \angle x v y $$	given
4 $$ \overline { v x } \cong \overline { v x } $$	definition of angle bisector
5	reflexive property of congruence

Explanation:

Step1: Recall Reflexive Property

The Reflexive Property of Congruence states that any segment or angle is congruent to itself. For triangles \( \triangle VXY \) and \( \triangle VXW \), the side \( VX \) is common to both triangles. But wait, looking at the reason for step 5, it's the Reflexive Property, and we need to identify the congruent parts. Wait, actually, the reflexive property for a side here would be \( \overline{VX} \cong \overline{VX} \), but no, wait, in the triangles \( \triangle VXY \) and \( \triangle VXW \), the common side is \( VX \), but also, maybe the side \( VX \) is shared, but let's check the triangles. Wait, the triangles are \( \triangle VXY \) and \( \triangle VXW \). So the sides: \( VX \) is common, so \( \overline{VX} \cong \overline{VX} \) by reflexive property. But wait, the statement for step 5, given the reason is Reflexive Property of Congruence, the statement should be a segment or angle congruent to itself. Wait, maybe I made a mistake. Wait, the triangles are \( \triangle VXY \) and \( \triangle VXW \). Let's list the parts:

- \( \angle W \cong \angle Y \) (given)
- \( \angle WVX \cong \angle XVY \) (since \( VX \) bisects \( \angle WVY \))
- And the side \( VX \) is common to both triangles, so \( \overline{VX} \cong \overline{VX} \) by reflexive property. Wait, but in the table, step 4 has \( \overline{VX} \cong \overline{VX} \) with reason Definition of angle bisector? No, wait, no, step 1: \( VX \) bisects \( \angle WVY \), so step 3: \( \angle WVX \cong \angle XVY \) (reason: definition of angle bisector). Step 4: maybe I misread. Wait, the problem is to complete step 5. The reason is Reflexive Property of Congruence, so the statement should be a segment or angle that is congruent to itself. In triangles \( \triangle VXY \) and \( \triangle VXW \), the side \( VX \) is common, but also, maybe the side \( VX \) is \( \overline{VX} \cong \overline{VX} \), but step 4 already has that? Wait, no, maybe the triangles have \( \overline{VX} \) as a common side, but the reflexive property here is for \( \overline{VX} \cong \overline{VX} \). Wait, no, perhaps the correct statement for step 5 is \( \overline{VX} \cong \overline{VX} \), but that's already in step 4? No, maybe I messed up. Wait, let's re-express:

To prove \( \triangle VXY \cong \triangle VXW \), we can use AAS (Angle-Angle-Side) or ASA. We have \( \angle Y \cong \angle W \) (given), \( \angle XVY \cong \angle WVX \) (since \( VX \) bisects \( \angle WVY \)), and the side \( VX \) is common, so by AAS, the triangles are congruent. The reflexive property is for the common side, so the statement for step 5 should be \( \overline{VX} \cong \overline{VX} \), but wait, the table's step 4: let's check the original table.

Original table:

1. \( \overleftrightarrow{VX} \) bisects \( \angle WVY \) - Reason: (empty)
2. \( \angle W \cong \angle Y \) - Reason: Given
3. \( \angle WVX \cong \angle XVY \) - Reason: Given (wait, no, reason should be definition of angle bisector)
4. \( \overline{VX} \cong \overline{VX} \) - Reason: Definition of angle bisector? No, that's wrong. Wait, the user's table has step 4: \( \overline{VX} \cong \overline{VX} \) with reason Definition of angle bisector, which is incorrect. But we need to complete step 5. The reason is Reflexive Property of Congruence, so the statement must be a segment or angle congruent to itself. In the two triangles \( \triangle VXY \) and \( \triangle VXW \), the side \( VX \) is common, so \( \overline{VX} \cong \overline{VX} \) by reflexive property. Wait, but step 4 already has that? No, maybe the table is misformatted. Alternatively, maybe the triangles have \( \overline{VX} \) as a common side, and the reflexive property is for \( \overline{VX} \cong \overline{VX} \). So the statement for step 5 is \( \overline{VX} \cong \overline{VX} \)? No, that can't be. Wait, maybe I made a mistake. Let's think again.

Wait, the triangles are \( \triangle VXY \) and \( \triangle VXW \). Let's list the corresponding parts:

- \( \angle Y \cong \angle W \) (given)
- \( \angle XVY \cong \angle WVX \) (since \( VX \) bisects \( \angle WVY \))
- \( \overline{VX} \cong \overline{VX} \) (common side, reflexive property)

So by AAS (Angle-Angle-Side), the triangles are congruent. So the third angle-side pair is the common side \( VX \), so the statement for step 5, with reason Reflexive Property of Congruence, is \( \overline{VX} \cong \overline{VX} \). Wait, but step 4 in the table has \( \overline{VX} \cong \overline{VX} \) with reason Definition of angle bisector, which is incorrect. But we need to complete step 5. Maybe the table has a typo, but according to the reflexive property, the statement should be \( \overline{VX} \cong \overline{VX} \).

Step2: Confirm the Statement

The Reflexive Property of Congruence states that a segment is congruent to itself. So for \( \triangle VXY \) and \( \triangle VXW \), the side \( \overline{VX} \) is common to both triangles, so \( \overline{VX} \cong \overline{VX} \) by the Reflexive Property of Congruence. Therefore, the statement for step 5 is \( \overline{VX} \cong \overline{VX} \).

Answer:

\( \overline{VX} \cong \overline{VX} \)