Question

t is the midpoint of \\( \overline { s u } \\). complete the proof that \\( \triangle t u w \cong \triangle t s v \\).

Explanation:

Step1: Definition of midpoint

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

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

We have \(\overline{UW}\cong\overline{SV}\) (given), \(\overline{TW}\cong\overline{TV}\) (given), and \(\overline{ST}\cong\overline{TU}\) (from step 1).
The SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
For \(\triangle TUW\) and \(\triangle TSV\), we have:

• \(UW = SV\) (given \(\overline{UW}\cong\overline{SV}\))
• \(TW=TV\) (given \(\overline{TW}\cong\overline{TV}\))
• \(TU = ST\) (from \(\overline{ST}\cong\overline{TU}\))

So, \(\triangle TUW\cong\triangle TSV\) by the SSS congruence criterion.

Answer:

1. Reason: Definition of midpoint
2. Reason: SSS (Side - Side - Side) congruence criterion