Question

w is the midpoint of \\( \overline{tv} \\) and \\( \overline{su} \\). complete the proof that \\( \triangle tuw \cong \triangle vsw \\).

statement	reason
2	w is the midpoint of \\( \overline{su} \\)	given
3	\\( \overline{tw} \cong \overline{vw} \\)
4	\\( \overline{sw} \cong \overline{uw} \\)
5	\\( \angle swv \cong \angle twu \\)
6	\\( \triangle tuw \cong \triangle vsw \\)	sas

Explanation:

Step1: Definition of midpoint

If \(W\) is the mid - point of \(\overline{TV}\), then by the definition of a mid - point (a mid - point of a line segment divides the line segment into two congruent parts), we have \(\overline{TW}\cong\overline{VW}\).

Step2: Definition of midpoint

If \(W\) is the mid - point of \(\overline{SU}\), then by the definition of a mid - point, we have \(\overline{SW}\cong\overline{UW}\).

Step3: Vertical angles

\(\angle SWV\) and \(\angle T WU\) are vertical angles. By the vertical angles theorem (vertical angles are congruent), \(\angle SWV\cong\angle T WU\).

Answer:

1. Definition of mid - point; 4. Definition of mid - point; 5. Vertical angles theorem.