Question

y is the midpoint of \\(\overline{wx}\\). complete the proof that \\(\triangle vxy \cong \triangle vwy\\).

statement	reason
2. \\(\overline{vw} \cong \overline{vx}\\)
3. \\(\overline{wy} \cong \overline{xy}\\)
4. \\(\overline{vy} \cong \overline{vy}\\)
5. \\(\triangle vxy \cong \triangle vwy\\)

Explanation:

Step1: Reason for Statement 2

From the diagram, the marks on \( \overline{VW} \) and \( \overline{VX} \) indicate they are congruent. So the reason is "Given (or Marked congruent segments)".

Step2: Reason for Statement 3

Since \( Y \) is the midpoint of \( \overline{WX} \), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So \( \overline{WY} \cong \overline{XY} \) because "Definition of midpoint".

Step3: Reason for Statement 4

\( \overline{VY} \) is a common side to both \( \triangle VXY \) and \( \triangle VWY \). So the reason is "Reflexive property of congruence (a segment is congruent to itself)".

Step4: Reason for Statement 5

We have \( \overline{VW} \cong \overline{VX} \) (Statement 2), \( \overline{WY} \cong \overline{XY} \) (Statement 3), and \( \overline{VY} \cong \overline{VY} \) (Statement 4). By the Side - Side - Side (SSS) Congruence Postulate, if three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. So \( \triangle VXY \cong \triangle VWY \) by SSS.

Answer:

1. Reason for Statement 1: Given
2. Reason for Statement 2: Marked congruent segments (or Given from diagram marks)
3. Reason for Statement 3: Definition of midpoint
4. Reason for Statement 4: Reflexive property of congruence
5. Reason for Statement 5: SSS Congruence Postulate