Question

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y is the midpoint of \\(\overline{xz}\\), \\(\triangle wmy\\) is equilateral, and \\(\overline{wx} \cong \overline{vz}\\). complete the proof that \\(\triangle wxy \cong \triangle vzy\\).

statement	reason
2 \\(\triangle wmy\\) is equilateral	given
3 \\(\overline{wx} \cong \overline{vz}\\)	given
4 \\(\overline{yz} \cong \overline{xy}\\)	definition of midpoint
5 \\(\overline{vy} \cong \overline{wy}\\)	definition of equilateral triangle
6

(image of a geometric figure with points x, w, v, z, y connected as per the problem context)

Explanation:

Step1: Analyze Given Info

We know \( Y \) is midpoint of \( \overline{XZ} \) (so \( XY = YZ \)), \( \triangle WMY \) is equilateral (so \( WY = MY = WM \)), \( \overline{WX} \cong \overline{VZ} \), and \( \overline{VY} \cong \overline{WY} \), \( \overline{YZ} \cong \overline{XY} \). We need to prove \( \triangle WXY \cong \triangle VZY \).

Step2: Identify Congruence Criteria

To prove \( \triangle WXY \cong \triangle VZY \), we can use SSS (Side - Side - Side) congruence. We have:

• \( XY = YZ \) (midpoint definition, step 4)
• \( WX = VZ \) (given, step 3)
• \( WY = VY \) (given, step 5)

So the statement for step 6 should be \( \triangle WXY \cong \triangle VZY \) and the reason is SSS (Side - Side - Side) Congruence Postulate.

Answer:

Statement: \( \triangle WXY \cong \triangle VZY \)
Reason: SSS (Side - Side - Side) Congruence Postulate