Question

x is the midpoint of \\( \overline{vw} \\) and \\( \overline{uv} \cong \overline{uw} \\). complete the proof that \\( \angle uxv \cong \angle uxw \\).

statement	reason
2	\\( \overline{uv} \cong \overline{uw} \\)
3	\\( \overline{vx} \cong \overline{wx} \\)
4	\\( \overline{ux} \cong \overline{ux} \\)
5	\\( \triangle uvx \cong \triangle uwx \\)
6	\\( \angle uxv \cong \angle uxw \\)

Explanation:

Step1: Given

Statement 1: \(X\) is the midpoint of \(\overline{VW}\) (Given).
Statement 2: \(\overline{UV}\cong\overline{UW}\) (Given).

Step2: Mid - point definition

Since \(X\) is the midpoint of \(\overline{VW}\), by the definition of a mid - point, \(\overline{VX}\cong\overline{WX}\) (If a point is the mid - point of a segment, then it divides the segment into two congruent segments).

Step3: Reflexive property

For any segment \(\overline{UX}\), \(\overline{UX}\cong\overline{UX}\) by the reflexive property of congruence (\(a\cong a\) for any geometric figure \(a\)).

Step4: SSS (Side - Side - Side) congruence criterion

In \(\triangle UVX\) and \(\triangle UWX\), we have \(\overline{UV}\cong\overline{UW}\) (given), \(\overline{VX}\cong\overline{WX}\) (from the mid - point definition), and \(\overline{UX}\cong\overline{UX}\) (reflexive property). So, \(\triangle UVX\cong\triangle UWX\) by the SSS (Side - Side - Side) congruence criterion.

Step5: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle UVX\cong\triangle UWX\), their corresponding parts are congruent. So, \(\angle UXV\cong\angle UXW\) by CPCTC.

Answer:

Statement	Reason
2. \(\overline{UV}\cong\overline{UW}\)	Given
3. \(\overline{VX}\cong\overline{WX}\)	Definition of mid - point
4. \(\overline{UX}\cong\overline{UX}\)	Reflexive property of congruence
5. \(\triangle UVX\cong\triangle UWX\)	SSS (Side - Side - Side) congruence criterion
6. \(\angle UXV\cong\angle UXW\)	CPCTC (Corresponding Parts of Congruent Triangles are Congruent)