Question

u is the midpoint of \\( \overline{tv} \\), \\( \overline{vx} \cong \overline{tw} \\), and \\( \overline{ux} \cong \overline{uw} \\). complete the proof that \\( \angle vux \cong \angle tuw \\).

Explanation:

Step1: Given

Statement 1, 2, 3 are given in the problem.

Step2: Definition of mid - point

Since \(U\) is the mid - point of \(\overline{TV}\), by the definition of a mid - point (a point that divides a segment into two congruent segments), we have \(\overline{TU}\cong\overline{UV}\).

Step3: SSS (Side - Side - Side) Congruence Criterion

We have \(\overline{TU}\cong\overline{UV}\), \(\overline{VX}\cong\overline{TW}\), \(\overline{UX}\cong\overline{UW}\). So, by the SSS (Side - Side - Side) congruence criterion (\(SSS\): If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent), \(\triangle TUW\cong\triangle VUX\).

Step4: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Since \(\triangle TUW\cong\triangle VUX\), by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), \(\angle VUX\cong\angle TUW\).

Answer:

1. Given
2. Given
3. Given
4. Definition of mid - point
5. \(SSS\) (Side - Side - Side)
6. \(CPCTC\) (Corresponding Parts of Congruent Triangles are Congruent)