Question

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

1 u is the midpoint of \\( \overline{tv} \\) given
2 \\( \overline{vx} \cong \overline{uw} \\) given
3 \\( \overline{tw} \cong \overline{ux} \\) given
4 \\( \overline{tu} \cong \overline{uv} \\)
5 \\( \triangle tuw \cong \triangle uvx \\)

Explanation:

Step1: Definition of midpoint

Since \(U\) is the midpoint of \(\overline{TV}\), by the definition of a midpoint, a midpoint divides a segment into two congruent segments. So, \(\overline{TU}\cong\overline{UV}\).

Step2: SSS (Side - Side - Side) Congruence Criterion

We are given \(\overline{VX}\cong\overline{UW}\), \(\overline{TW}\cong\overline{UX}\), and from Step 1 we have \(\overline{TU}\cong\overline{UV}\).
In \(\triangle TUW\) and \(\triangle UVX\), we have three pairs of congruent sides:

• \(TU = UV\) (from Step 1)
• \(UW=VX\) (given)
• \(TW = UX\) (given)

By the SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. So, \(\triangle TUW\cong\triangle UVX\)

Answer:

1. Reason: Definition of mid - point; 5. Reason: SSS (Side - Side - Side) congruence criterion.